Racial inequality in the United States

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In the United States, racial inequality refers to the social inequality and advantages and disparities that affect different races. These can also be seen as a result of historic oppression, inequality of inheritance, or racism and prejudice, especially against minority groups.

There are vast differences in wealth across racial groups in the United States. The wealth gap between Caucasian and African American families substantially increased from $85,000 in 1984 to $236,500 in 2009. There are many causes that relate to racial inequality such as: Years of home ownership, household income, unemployment, education, and inheritance.

Under slavery, African Americans were treated as property. After the American Civil War, Black sharecroppers became trapped in debt. African Americans were rarely able to homestead. The Freedman's Savings Bank failed, losing many Black assets.

Exclusions from Social Security disproportionately affected African Americans. Savings were spent for retirement instead of handed down as inheritance. African Americans are less likely to receive inheritance and more likely to aid poor family members.

The Federal Housing Administration and Veteran's Administration shut out African Americans by giving loans to suburbs instead of central cities. Housing segregation caused unequal living standards and poverty. Public education greatly relies on local property taxes, with racial inequality between White affluent suburbs and poor minorities in inner-cities.

Criminal records lead to employment and income struggles. Inability to make bail and quality counsel are factors. Racial segregation and racial profiling lead to differences between races.

Definitions

In social science, racial inequality is typically defined as "imbalances in the distribution of power, economic resources, and opportunities." Racial inequalities have manifested in American society in ways ranging from racial disparities in wealth, poverty rates, bankruptcy, housing patterns, educational opportunities, unemployment rates, and incarceration rates. Current racial inequalities in the U.S. have their roots in over 300 years of cultural, economic, physical, legal, and political discrimination based on race.

Racial wealth gap

A study by the Brandeis University Institute on Assets and Social Policy which followed the same sets of families for 25 years found that there are vast differences in wealth across racial groups in the United States. The wealth gap between Caucasian and African-American families studied nearly tripled, from $85,000 in 1984 to $236,500 in 2009. The study concluded that factors contributing to the inequality included years of home ownership (27%), household income (20%), education (5%), and familial financial support and/or inheritance (5%). In an analysis of the American Opportunity Accounts Act, a bill to introduce Baby Bonds, Morningstar reported that by 2019 white families had more than seven times the wealth of the average Black family, according to the Survey of Consumer Finances.

Median household income along ethnic lines in the United States

Wealth can be defined as "the total value of things families own minus their debts." In contrast, income can be defined as, "earnings from work, interest and dividends, pensions, and transfer payments." Wealth is an important factor in determining the quality of both individual and family life chances because it can be used as a tool to secure a desired quality of life or class status and enables individuals who possess it to pass their class status to their children. Family inheritance, which is passed down from generation to generation, helps with wealth accumulation. Wealth can also serve as a safety net against fluctuations in income and poverty.

There is a large gap between the wealth of minority households and white households within the United States. The Pew Research Center's analysis of 2009 government data says the median wealth of white households is 20 times that of Black households and 18 times that of Hispanic households. In 2009 the typical Black household had $5,677 in wealth, the typical Hispanic had $6,325, and the typical white household had $113,149. Furthermore, 35% of African American and 31% of Hispanic households had zero or negative net worth in 2009 compared to 15% of white households. While in 2005 median Asian household wealth was greater than white households at $168,103, by 2009 that changed when their net worth fell 54% to $78,066, partially due to the arrival of new Asian immigrants since 2004; not including newly arrived immigrants, Asian net wealth only dropped 31%. As shown on "EURweb – Electronic Urban Report" According to the Federal Reserve Survey of Consumer Finances, of the 14 million Black households, only 5% have more than $350,000 in net worth while nearly 30% of white families have more than this amount. Less than 1% of Black families have over a million in net assets. while nearly 10% of white households, totaling over 8 million families have more than 1.3 million in net worth. According to the Federal Reserve of Cleveland the wealth gap between white and Black Americans has remained roughly the same since 1962, when the average white family had seven times the wealth of the average Black family.

Lusardi states that African Americans and Hispanics are more likely to face means-tested programs that discourage asset possession due to higher poverty rates. One-fourth of African Americans and Hispanics approach retirement with less than $1,000 net worth (without considering pensions and Social Security). Lower financial literacy is correlated with poor savings and adjustment behavior. Education is a strong predictor for wealth. One-fourth of African Americans and Hispanics that have less than a high school education have no wealth, but even with increased education, large differences in wealth remain.

Conley believes that the cause of Black-White wealth inequality may be related to economic circumstances and poverty because the economic disadvantages of African Americans can be effective in harming efforts to accumulate wealth. However, there is a five times greater chance of downward mobility from the top quartile to the bottom quartile for African Americans than there is for white Americans; correspondingly, African Americans rise to the top quartile from the bottom quartile at half the rate of white Americans. Bowles and Gintis conclude from this information that successful African Americans do not transfer the factors for their success as effectively as white Americans do. Other factors to consider in the recent widening of the minority wealth gap are the subprime mortgage crisis and financial crisis of 2007–2008. The Pew Research Center found that plummeting house values were the main cause of the wealth change from 2005 to 2009. Hispanics were hit the hardest by the housing market meltdown possibly because a disproportionate share of Hispanics live in California, Florida, Nevada, and Arizona, which are among the states with the steepest declines in housing values. From 2005 to 2009 Hispanic homeowners' home equity declined by Half, from $99,983 to $49,145, with the homeownership rate decreasing by 4% to 47%. A 2015 Measure of America study commissioned by the ACLU on the long-term consequences of discriminatory lending practices found that the financial crisis will likely widen the Black-white wealth gap for the next generation.

The racial wealth gap essentially is composed of a private wealth management industry maintaining Whiteness to act as a barrier to prevent those of color from equal financial development. This disparity has been debated, but never disputed due to its "very real" implications it has on African Americans. Data has shown that "among racial and ethnic groups, African Americans had the highest poverty rate at 27.4%”.

History

Before the 1808 abolition of the transatlantic slave trade, Africans would be captured and brought into the United States as enslaved people, depriving them of all property, and in some cases family. In order to prevent rebellion or escape, the slave codes in some states banned education of slaves, especially teaching a slave to read or write. Redistribution of land from white owners to the people formerly forced to work it was attempted under the forty acres and a mule policy of Union General William Tecumseh Sherman. This was reversed by President Andrew Johnson, a Southern Democrat who also opposed political rights for African Americans and protections against white violence in the South. Slavery continued in the border states until ratification of the Thirteenth Amendment to the United States Constitution in December 1865.

The Freedmen's Bureau was created as part of the War Department by President Abraham Lincoln to provide shelter and supplies to freed slaves. It was supported by the Republican Congress over the veto of Andrew Johnson, but was soon de-funded and abandoned by a Democrat-controlled Congress in 1872.

While free African Americans owned around $50 million by 1860, farm tenancy and sharecropping replaced slavery after the American Civil War because newly freed African American farmers did not own land or supplies and had to depend on the White Americans who rented the land and supplies out to them. At the same time, southern Blacks were trapped in debt and denied banking services while White citizens were given low interest loans to set up farms in the Midwest and Western United States. White homesteaders were able to go West and obtain unclaimed land through government grants, while the land grants and rights of African Americans were rarely enforced.

After the Civil War the Freedman's Bank helped to foster wealth accumulation for African Americans. However, it failed in 1874, partially because of suspicious high-risk loans to White banks and the Panic of 1873. This lowered the support African Americans had to open businesses and acquire wealth. In addition, after the bank failed, taking the assets of many African Americans with it, many African Americans did not trust banks. There was also the threat of lynching to any African American who achieved success.

In addition, when Social Security was first created during the Great Depression, it exempted agricultural and domestic workers, which disproportionately affected African Americans and Hispanics. Consequently, the savings of retired or disabled African Americans was spent during old age instead of handed down and households had to support poor elderly family members. The Homeowner's Loan Corporation that helped homeowners during the Great Depression gave African American neighborhoods the lowest rating, ensuring that they defaulted at greater rates than White Americans. The Federal Housing Authority (FHA) and Veteran's Administration (VA) shut out African Americans by giving loans to suburbs instead of central cities after they were first founded.

Inheritance and parental financial assistance

Bowman states that "in the United States, the most significant aspect of multigenerational wealth distribution comes in the forms of gifts and inheritances." However, the multigenerational absence of wealth and asset attainment for African Americans makes it almost impossible for them to make significant contributions of wealth to the next generation. Data shows that financial inheritances could account for 10 to 20 percent of the difference between African American and White American household wealth.

Using the Health and Retirement Study (HRS) of 1992 Avery and Rendall estimated that only around one-tenth of African Americans reported receiving inheritances or substantial inter vivo transfers ($5,000 or more) compared to one-third of white Americans. In addition, the 1989 Survey of Consumer Finances (SCF) reported that the mean and median values of those money transfers were significantly higher for white American households: the mean was $148,578 households compared to $85,598 for African American households and the median was $58,839 to $42,478. The large differences in wealth in the parent-generations were a dominant factor in prediction the differences between African American and white American prospective inheritances. Avery and Rendall used 1989 SCF data to discover that the mean value in 2002 of white Americans' inheritances was 5.46 times that of African Americans', compared to 3.65 that of current wealth. White Americans received a mean of $28,177 that accounted for 20.7% of their mean wealth while African Americans received a mean of $5,165 that accounted for 13.9% of their mean current wealth. Non-inherited wealth was more equally distributed than inherited wealth.

Avery and Rendall found that family attributes favored white Americans when it came to factor influencing the amount of inheritances. African Americans were 7.3% less likely to have live parents, 24.5% more likely to have three or more siblings, and 30.6% less likely to be married or cohabiting (meaning there are two people who could gain inheritances to contribute to the household) Keister discovered that large family size has a negative effect on wealth accumulation. These negative effects are worse for the poor and African Americans and Hispanics are more likely to be poor and have large families. More children also decrease the amount of gifts parents can give and the inheritance they leave behind for the children.

Angel's research into inheritance showed that older Mexican American parents may give less financial assistance to their children than non-Hispanic White Americans because of their relatively high fertility rate so children have to compete for the available money. There are studies that indicate that elderly Hispanic parents of all backgrounds live with their adult children due to poverty and would choose to do otherwise, even if they had the resources to do so. African American and Latino families are less likely to financially aid adult children than non-Hispanic white families.

Income effects

See also: Racial wage gap in the United States

The racial wealth gap is visible in terms of dollar for dollar wage and wealth comparisons. For example, middle-class Blacks earn seventy cents for every dollar earned by similar middle-class whites. Race can be seen as the "strongest predictor" of one's wealth.

Krivo and Kaufman found that information supporting the fact that increases in income does not affect wealth as much for minorities as it does for white Americans. For example, a $10,000 increase in income for white Americans increases their home equity $17,770 while the same increase only increase the home equities for Asians by $9,500, Hispanics by $15,150, and African Americans by $15,900.

Financial decisions

Investments

Conley states that differences between African American and white American wealth begin because people with higher asset levels can take advantage of riskier investment instruments with higher rates of returns. Unstable income flows may lead to "cashing in" of assets or accumulation of debt over time, even if the time-averaged streams of income and savings are the same. African Americans may be less likely to invest in the stock market because they have a smaller parental head-start and safety net.

Chong, Phillips and Phillips state that African Americans, Hispanics, and Asians invest less heavily in stocks than white Americans. Hispanics and in some ways African Americans accumulate wealth slower than white Americans because of preference for near-term saving, favoring liquidity and low investment risk at the expense of higher yielding assets. These preferences may be due to low financial literacy leading to a lack of demand for investment services. According to Lusardi, even though the stock market increased in value in the 1990s, only 6-7% of African Americans and Hispanics held stocks, so they did not benefit as much from the value increase.

Use of financial services

See also: Criticism of credit scoring systems in the United States

The Federal Deposit Insurance Corporation in 2009 found that 7.7% of United States households are unbanked. Minorities are more likely than white Americans to not have a banking account. 3.5% of Asians, 3.3% of white Americans, 21.7% of African Americans and 19.3% of Hispanics and 15.6% of remaining racial/ethnic categories do not have banking accounts.

Lusardi's research revealed that education increases one's chances of having a banking account. A full high school education increases the chance of having a checking account by 15% compared to only an elementary education; having a parent with a high school education rather than only an elementary education increases one's chances of having a checking account by 2.8%. This difference in education level may explain the large proportion of "unbanked" Hispanics. The 2002 National Longitudinal Survey found that while only 3% of white Americans and 4% of African Americans had only an elementary education, close to 20% of Hispanics did and 43% of Hispanics had less than a high school education Ibarra and Rodriguez believe that another factor that influences the Hispanic use of banking accounts is credit. Latinos are also more likely than white Americans or African Americans to have no or a thin credit history: 22% of Latinos have no credit score in comparison to 4% of white Americans and 3% of African Americans.

Not taking other variables into account, Chong, Phillips, and Phillips survey of zip codes found that minority neighborhoods don't have the same access to financial planning services as white neighborhoods. There is also client segregation by investable assets. More than 80% of financial advisors prefer that clients have at least $100,000 in investable assets and more than 50% have a minimum asset requirement of $500,000 or above. Because of this, financial planning is possibly beyond the reach of those with low income, which comprises a large portion of African-Americans and Hispanics. Fear of discrimination is another possible factor. Minorities may be distrustful of banks and lack of trust was commonly reported as why minorities, people with low education, and the poor chose not to have banking accounts. Evidence suggests that women of color are disproportionately likely to plan on using informal borrowing as their sole strategy for coping with an emergency expense, potentially due to lack of access to formal banking services.

Aid to family members

Though African Americans who attend college get a similar boost in income to white peers, they tend not to have a similar increase in wealth, largely because they spend more money helping poor family members, including older relatives who in white families are instead more likely to help younger kin.

Health care

Main articles: Race and health in the United States, Social determinants of mental health, and Social determinants of health

Black Americans face consistently worse health outcomes than white, Asian, and Hispanic Americans. Black women are 2½ times more likely to die of maternal causes than white women and this rate increases to 3 times when compared to Hispanic Americans. The infant mortality rate for Black Americans is 11 per 1,000 births which is higher than the US average of 5.7. There exists gaps in life expectancy between races with Black and Native Americans having the lowest life expectancies. The gap between Black and white Americans on average is four years; however, there is great variation between states and even on smaller levels. For example in Wisconsin this gap is six years, and in Washington, D.C., this gap is more than ten years. African American women have the highest rate of obesity or being overweight in the US and non-Hispanic Blacks are 1.3 times more likely to be obese than non-Hispanic Whites.

Poverty

There are large differences in poverty rates across racial groups. In 2009, the poverty rate across the nation was 9.9%. This data illustrates that Hispanics and Blacks experience disproportionately high percentages of poverty in comparison to non-Hispanics whites and Asians. In discussing poverty, it is important to distinguish between episodic poverty and chronic poverty.

Episodic poverty

The U.S. Census Bureau defines episodic poverty as living in poverty for less than 36 consecutive months. From the period between 2004 and 2006 the episodic poverty rate was 22.6% for non-Hispanic whites, 44.5% for Blacks, and 45.8% for Hispanics. Blacks and Hispanics experience rates of episodic poverty that are nearly double the rates of non-Hispanic whites.

Chronic poverty

The U.S. Census Bureau defines chronic poverty as living in poverty for 36 or more consecutive months. From the period between 2004 and 2006 the chronic poverty rate was 1.4% for non-Hispanic whites, 4.5% for Hispanics, and 8.4% for Blacks. Hispanics and Blacks experience much higher rates of chronic poverty when compared to non-Hispanic whites.

Length of poverty spell

The U.S. Census Bureau defines length of poverty spell as the number of months spent in poverty. The median length of poverty spells was 4 months for non-Hispanic whites, 5.9 months for Blacks, and 6.2 months for Hispanics. The length of time spent in poverty varies by race. Non-Hispanic whites experience the shortest length of poverty spells when compared to Blacks and Hispanics.

Housing

Housing segregation

Main article: Housing segregation in the United States

Housing segregation in the United States is the practice of denying African American or other minority groups equal access to housing through the process of misinformation, denial of realty and financing services, and racial steering. Housing policy in the United States has influenced housing segregation trends throughout history. Key legislation include the National Housing Act of 1934, the GI Bill, and the Fair Housing Act. Factors such as socioeconomic status, spatial assimilation, and immigration contribute to perpetuating housing segregation. The effects of housing segregation include relocation, unequal living standards, and poverty. However, there have been initiatives to combat housing segregation, such as the Section 8 housing program.

Racial residential segregation doubled from 1880 to 1940. Southern urban areas were the most segregated. Segregation was highly correlated with lynchings of African-Americans. Segregation adversely affected both black and white homeownership rates, as well as caused higher crime rates. Areas with housing segregation had worse health outcomes for both whites and Blacks. Residential segregation accounts for a substantial share of the Black-white gap in birth weight. Segregation reduced upward economic mobility.

White communities are more likely to have strict land use regulations (and whites are more likely to support those regulations). Strict land use regulations are an important driver of housing segregation along racial lines in the United States.

Eviction

Black tenants face significantly higher filing and eviction rates than their white counterparts. Looking at neighborhood racial composition in Milwaukee, sociologist Matthew Desmond found that majority-black neighborhoods had an average annual eviction rate of 7.4%, compared to 1.4% in majority-white neighborhoods. In this study, Desmond also emphasizes the dual disadvantage black women face in housing—black women face the highest eviction rates of any demographic group. In an interview with The Atlantic, Desmond reported that approximately one in five black women will experience eviction, compared to one in fifteen white women. Eviction rates are also linked to the racial concentration of neighborhoods. The RVA Eviction Lab, in Richmond, Virginia, estimates that as the proportion of a neighborhood's black population increases by 10%, eviction rates would increase by 1.2%.

Hispanic renters also face higher filing and eviction rates than their white counterparts. In a study published in the Harvard Civil Rights–Civil Liberties Law Review, researchers investigated the relationship between Hispanic origin and eviction in Milwaukee. These researchers saw a strong correlation between Hispanic tenants' risk of eviction and neighborhood racial composition. In Milwaukee neighborhoods that were two-thirds white, approximately 80% of landlords were white. In these same neighborhoods, the average eviction rate was 25%, yet the eviction rate for Hispanics was upwards of 35%. The study also found that Hispanic renters were significantly more likely to be evicted by white landlords than non-white landlords. According to Greenberg et al, these findings suggest that discrimination contributes to racial disparities in Milwaukee eviction rates.

Education

Main article: Educational inequality in the United States

See also: Racial achievement gap in the United States

College attendance, analyzed by race and schools' overall admission rates. Shown by comparative areas of upper four pie charts, elite schools make up a small fraction of all enrollment.

In the United States, funding for public education relies greatly on local property taxes. Local property tax revenues may vary between different neighborhoods and school districts. This variance of property tax revenues amongst neighborhoods and school districts leads to inequality in education. This inequality manifests in the form of available school financial resources which provide educational opportunities, facilities, and programs to students. For every student enrolled, the average nonwhite school district receives $2,226 less than a white school district.

Returning to the concept of residential segregation, it is known that affluence and poverty have become both highly segregated and concentrated in relation to race and location. Residential segregation and poverty concentration is most markedly seen in the comparison between urban and suburban populations in which suburbs consist of majority white populations and inner-cities consist of majority minority populations. According to Barnhouse-Walters (2001), the concentration of poor minority populations in inner-cities and the concentration of affluent white populations in the suburbs, "is the main mechanism by which racial inequality in educational resources is reproduced."

In August 2020, the US Justice Department argued that Yale University discriminated against Asian candidates on the basis of their race, a charge the university denied.

Unemployment rates

In 2016, the unemployment rate was 3.8% for Asians, 4.6% for non-Hispanic whites, 6.1% for Hispanics, and 9.0% for Blacks, all over the age of 16. In terms of unemployment, it can be seen that there are two-tiers: relatively low unemployment for Asians and whites, relatively high unemployment for Hispanics and Blacks.

Potential explanations

Several theories have been offered to explain the large racial gap in unemployment rates:

Segregation and job decentralization

This theory argues that the effects of racial segregation pushed Blacks and Hispanics into the central city during a time period in which jobs and opportunities moved to the suburbs. This led to geographic separation between minorities and job opportunities which was compounded by struggles to commute to jobs in the suburbs due to lack of means of transportation. This ultimately led to high unemployment rates among minorities.

White gains

This theory argues that the reason minority disadvantage exists is because the majority group is able to benefit from it. For example, in terms of the labor force, each job not taken by a Black person could be job that gets occupied by a white person. This theory is based on the view that the white population has the most to gain from the discrimination of minority groups. In areas where there are large minority groups, this view predicts high levels of discrimination to occur for the reason that white populations stand to gain the most in those situations.

Job skill differentials

This theory argues that the unemployment disparity can be attributed to lower rates of academic success among minority groups (especially Black Americans) leading to a lack of skills necessary for entering the modern work force. The author remains unclear why Black Americans have low academic success.

Other explanations

It is politically incorrect to assume that racial inequality is caused by differences in skills or preferences. The lack of open discussion leads to ethnic groups being treated equally. This means that diverse groups receive the same offers which have different advantages for different groups, which further increases inequality.

Crime rates and incarceration

Main articles: Race and crime in the United States and Race in the United States criminal justice system

In 2008, the prison population under federal and state correctional jurisdiction was over 1,610,446 prisoners. Of these prisoners, 20% were Hispanic (compared to 16.3% of the U.S. population that is Hispanic), 34% were White (compared to 63.7% of the U.S. population that is White), and 38% were Black (compared to 12.6% of the U.S. population that is Black). Additionally, Black males were imprisoned at a rate 6.5 times higher than that of their White male counterparts. According to a 2012 study by the U.S. Census Bureau, "over half the inmates incarcerated in our nation's jails is either Black or Hispanic." According to a report by the National Council of La Raza, research obstacles undermine the census of Latinos in prison, and "Latinos in the criminal justice system are seriously undercounted. The true extent of the overrepresentation of Latinos in the system probably is significantly greater than researchers have been able to document.

Consequences of a criminal record

The injustices of a criminal justice system disproportionately impact Black people; maintaining these racial disparities has a high cost for individuals, families, and communities. On an individual level, a criminal conviction may equate to loss of access to employment, housing, and public service opportunities. On the community level, the dipropionate incarceration of people from poorer communities depletes the economic resources of said community and curates cycles of poverty that becomes increasingly harder to get rid of. These communities also face increased criminal justice involvement in their communities, making criminal justice contact a norm in the lives of an immense number of Black Americans.

After being released from prison, the consequences of having a criminal record are immense. Over 40 percent who are released will return to prison within the next few years. Those with criminal records who do not return to prison face significant struggles to find quality employment and income outcomes compared to those who do not have criminal records.

Those racially disparate employment consequences can arise from other forms of carceral contact, too, and they can have spillover effects on local communities. At the county level, for example, jail rather than prison incarceration has been found to significantly diminish local labor markets in areas with relatively high proportions of Black residents. 

Potential causes

Poverty

A potential cause of such disproportionately high incarceration rates for Black Americans is that Black Americans are disproportionately poor. Conviction is a crucial part of the process that leads to either guilt or innocence. There are two important factors that play a role in this part of the process: the ability to make bail and the ability to access high-quality legal counsel. Due to the fact that both of these important factors cost money, it is unlikely that poor Black Americans are able to afford them and benefit from them. Sentencing is another crucial part of the process that determines how long individuals will remain incarcerated. Several sociological studies have found that poor offenders receive longer sentences for violent crimes and crimes involving drug use, unemployed offenders are more likely to be incarcerated than their employed counterparts, and then even with similar crimes and criminal records minorities were imprisoned more often than whites.

Racial profiling

Racial profiling is defined as "any police-initiated action that relies on the race, ethnicity, or national origin, rather than the behavior of an individual or information that leads the police to a particular individual who has been identified as being, or having been, engaged in criminal activity." Another potential cause for the disproportionately high incarceration rates of Blacks and Hispanics is that racial profiling occurs at higher rates for Blacks and Hispanics. Eduardo Bonilla-Silva states that racial profiling can perhaps explain the over representation of Blacks and Hispanics in U.S. prisons. According to Michael L. Birzer, professor of criminal justice at Wichita State University and director of its School of Community Affairs, "racial minorities, particularly African Americans, have had a long and troubled history of disparate treatment by United States Criminal Justice Authorities." A report by the National Registry of Exonerations found that African Americans were seven times more likely to be falsely convicted compared to White Americans.

Racial segregation

"Racial residential segregation is a fundamental cause of racial disparities in health". Racial segregation can result in decreased opportunities for minority groups in income, education, etc. While there are laws against racial segregation, study conducted by D. R. Williams and C. Collins focuses primarily on the impacts of racial segregation, which leads to differences between races.

Police brutality

Main article: Police brutality in the United States

Significant racial discrepancies have been reported in the United States involving police brutality. Police brutality in the United States is defined as "the unwarranted or excessive and often illegal use of force against civilians by U.S. police officers." It can come in the form of murder, assault, mayhem, or torture, as well as less physical means of violence including general harassment, verbal abuse, and intimidation. it has been argued that the origins of racial inequality by way of police brutality in America date to colonial times when slavery was legal and widespread. Due to fear of slave revolts, White Americans began to organize groups of vigilantes who would use force to keep slaves from rebelling against their owners or escaping.

During the civil rights era, the existence of the racial disparities surrounding police brutality became more discussed. During peaceful protests for civil rights, some police would use tactics such as police dogs or fire hoses to control the protesters. In 1991, video footage was released of cab driver Rodney King being hit over 50 times by multiple police with their batons. The police were later acquitted for their actions which resulted in the 1992 Los Angeles riots. Allegations of police brutality continue to plague American police. An alleged example of police brutality includes Philando Castile, a 32-year-old Black male who was pulled over for a broken taillight. After being told by officer Yanez, to take out his license and insurance, Castile told the officer he had a firearm and that he was reaching into his pocket to get his wallet. Seconds later he was shot and killed by Yanez, who claimed he believed Castile was pulling his gun out. Yanez was charged with manslaughter and acquitted at trial.

Black Americans are 2–3 times more likely to be killed in a police shooting than White Americans and are more likely to be unarmed during those fatal instances. A study done by Joshua Correll at the University of Chicago shows what is called "The police officers dilemma", by setting up a video game in which police are given scenarios involving both Black and White men holding either a gun or non-threatening objects such as cellphones. The study found that armed Black men were shot more frequently than armed White men and were also shot more quickly. The police would also sometimes mistakenly shoot the unarmed Black targets, while neglecting to shoot the armed White targets. Militarized police units are more often deployed in Black neighborhoods even after adjusting for crime rates.

A 2020 study by Cody Ross et al. concluded that there was evidence of bias in police shootings of unarmed Blacks and that even when using crime as a benchmark "there is strong and statistically reliable evidence of anti-Black racial disparities in the killing of unarmed Americans by police". A 2019 study by Cesario et al. published in Social Psychological and Personality Science found that after adjusting for crime, there was "no systematic evidence of anti-Black disparities in fatal shootings, fatal shootings of unarmed citizens, or fatal shootings involving misidentification of harmless objects. A study by Harvard economist Roland Fryer found that for officer-involved shootings there were no racial disparities "in either the raw data or when contextual factors are taken into account". According to data from the Chicago Police Department police used more force against Black people than any other race despite the fact that they were less likely to resist arrest than Whites. Around 20% of the population of Minneapolis is Black but they are subjected to nearly 60% of total police use of force.

Color-blind racism

It is hypothesized by some scholars, such as Michelle Alexander, that in the since the Civil Rights Era, the United States has now switched to a new form of racism known as color-blind racism. Color-blind racism refers to "contemporary racial inequality as the outcome of nonracial dynamics."

The types of practices that take place under color blind racism are "subtle, institutional, and apparently nonracial." Those practices are not racially overt in nature such as racism under slavery, segregation, and Jim Crow laws. Instead, color-blind racism flourishes on the idea that race is no longer an issue in the country and that there are non-racial explanations for the state of inequality. Eduardo Bonilla-Silva writes that there are four frames of color-blind racism that support that view:

1. Abstract liberalism uses ideas associated with political liberalism. This frame is based in liberal ideas such as equal opportunity, individualism, and choice. It uses these ideas as a basis to explain inequality.
2. Naturalization explains racial inequality as a cause of natural occurrences. It claims that segregation is not the result of racial dynamics. Instead, it is the result of the naturally-occurring phenomena of individuals choosing likeness as their preference.
3. Cultural racism explains racial inequality through culture. Under this frame, racial inequalities are described as the result of stereotypical behavior of minorities. Stereotypical behavior includes qualities such as laziness and teenage pregnancy.
4. Minimization of racism attempts to minimize the factor of race as a major influence in affecting the life chances of minorities. It writes off instances and situations that could be perceived as discrimination to be hypersensitivity to the topic of race.

Natural disasters

Main article: Environmental racism in the United States

When a disaster strikes—be it a hurricane, tornado, or fire—some people are inherently more prepared than others. "While all members of populations are affected by disasters, research findings show that racial and ethnic minorities are less likely to evacuate and more affected by disasters" than their Caucasian counterparts. "During Hurricane Katrina, the large number of people seeking safety in designated shelters were disproportionately Black. In addition, the mortality rate for Blacks was 1.7 to 4 times higher than that of whites for all people ≥ 18." After Hurricane Katrina, many African Americans felt abandoned by the United States Government. 66% of African Americans "said that 'the government's response to [Katrina] would have been faster if most of the victims had been white.'" For a disproportionate share of the impoverished in New Orleans, many had, and continue to have, a difficult time preparing for storms. Factors such as, "cultural ignorance, ethnic insensitivity, racial isolation and racial bias in housing, information dissemination, and relief assistance" all greatly contribute to the disparities in disaster preparedness.

Credit scores

Credit score systems are well known to contain racial bias and have been shown to increase racial disparities as studies show that African American and American Latino populations have substantially lower scores than the white American population on average. Racial discrimination also results in impacts on the credit scores and economic security of communities of color—that ultimately, "entrenches and reinforces inequality by dictating a consumer's access to future opportunities".

Numerous studies have found racial disparities in credit scoring:

• 1996 study found African-Americans were three times as likely to have FICO scores below 620 as whites and that Hispanics were twice as likely.
• 1997 study found Black, Indigenous, and people of color [BIPOC] neighborhood consumers had lower credit scores.
• 2004 study found high Black, Indigenous, and people of color [BIPOC] zip codes to have significantly worse scores than non-Black, Indigenous, and people of color [BIPOC] zip codes.
• 2004 study found that African American and Hispanic consumers constituted over 60% of the consumers having the worst credit scores.
• 2004 study found the median credit score for whites in 2001 was 738, but the median credit score for African Americans was 676 and for Hispanics was 670.
• 2004 research study found fewer than 40% of consumers who lived in high-Black Indigenous and people of color [BIPOC] neighborhoods had credit scores of over 701.
• 2006 studied US counties with high Black, Indigenous, and people of color [BIPOC] populations determining that those countries had lower average credit scores than predominantly white counties.
• 2007 study by the Federal Trade Commission found that African Americans and Hispanics strongly overrepresented in the lowest scoring categories regarding auto insurance company's use of credit scores.
• 2007 report found significant racial disparities in 300,000 credit files matched with Social Security records with African American scores being half that of white, non-Hispanics.
• 2010 study found that African American in Illinois zip codes had scores of less than 620 at a rate of 54.2%. In zip codes that were majority Latino, 31.4% of individuals had a credit score of less than 620, and only 47.3% had credit scores greater than 700.
• 2012 study examined the credit scores for about 200,000 consumers finding the median FICO score in majority minority zip codes was in the 34th percentile, while it was in the 52nd percentile for low minority zip codes.
• 2023 expert report and study by Credlocity titled: "The Dark Side of Credit Scores: How Racial Bias and Injustice Affect Millions of Americans", found that the major credit bureaus are giving greater weight to the disputes submitted by white Americans than by Black, brown and Hispanic Americans, citing several whistleblower reports, lawsuits, and public studies.

The outcomes for Black Americans because of this bias are higher interest rates on home loans and auto loans; longer loan terms; increased debt collection default lawsuits, and an increase in the use of predatory lenders. FICO has defended the system stating that income, property, education, and employment are not evenly distributed across society and it is irrational to think an objective measure would not exhibit these discrepancies. Tamara Nopper, sociologist at The Center for Critical Race & Digital Studies has stated that to solve the true issue of racism is not just to regulate it, as politics focus on, but to eliminate it in favor of public-owned banks that serve the community instead of shareholders.

A related concept of insurance scoring has also been shown to discriminate along racial lines, disproportionately harming black and Latino populations.

Racial color blindness

From Wikipedia, the free encyclopedia

Racial color blindness refers to the belief that a person's race or ethnicity should not influence their legal or social treatment in society.

The multicultural psychology field generates four beliefs that constitute the racial color-blindness approach. The four beliefs are as follows: (1) skin color is superficial and irrelevant to the quality of a person's character, ability or worthiness, (2) in a merit-based society, skin color is irrelevant to merit judgments and calculation of fairness, (3) as a corollary, in a merit-based society, merit and fairness are flawed if skin color is taken into the calculation, (4) ignoring skin color when interacting with people is the best way to avoid racial discrimination.

The term metaphorically references the medical phenomenon of color blindness. Psychologists and sociologists also study racial color blindness. This is further divided into two dimensions, color evasion and power evasion. Color evasion is the belief that people should not be treated differently on the basis of their color. Power evasion posits that systemic advantage based on color should have no influence on what people can accomplish, and accomplishments are instead based solely on one's own work performance.

At various times in Western history, this term has been used to signal a desired or allegedly achieved state of freedom from racial prejudice or a desire that policies and laws should not consider race. Proponents of racial color blindness often assert that policies that differentiate by racial classification could tend to create, perpetuate or exacerbate racial divisiveness. Critics often believe it fails to address systemic discrimination.

It has been used by justices of the United States Supreme Court in several opinions relating to racial equality and social equity, particularly in public education.

In U.S. Supreme Court opinions

In his dissenting opinion to Plessy v. Ferguson (1896), Justice John Marshall Harlan wrote that "Our Constitution is color-blind, and neither knows nor tolerates classes among citizens. In respect of civil rights, all citizens are equal before the law. The humblest is the peer of the most powerful. The law regards man as man, and takes no account of his surroundings or of his color when his civil rights as guaranteed by the supreme law of the land are involved." His opinion could thus be interpreted as saying that laws should not differentiate between people of different races. His opinion was not the majority-supported decision, which at the time was that laws requiring racial segregation were allowable, establishing the idea that "separate but equal" treatment was constitutionally acceptable.

More recently, the term color blind has appeared in United States Supreme Court opinions involving affirmative action, in opinions that support consideration of race when evaluating laws and their effects:

• In a concurring opinion of Regents v. Bakke (1978), Justices William J. Brennan Jr., Byron White, Thurgood Marshall, and Harry Blackmun objected to the color blind term, writing that "we cannot ... let color blindness become myopia which masks the reality that many 'created equal' have been treated within our lifetimes as inferior both by the law and by their fellow citizens."
• In her dissenting opinion to Gratz v. Bollinger (2003), Justice Ruth Bader Ginsburg quoted from a 1966 5th Circuit decision: "'The Constitution is both color blind and color conscious. To avoid conflict with the equal protection clause, a classification that denies a benefit, causes harm, or imposes a burden must not be based on race. In that sense, the Constitution is color blind. But the Constitution is color conscious to prevent discrimination being perpetuated and to undo the effects of past discrimination.'"
• In his concurring opinion to PICS v. Seattle (2007), Justice Clarence Thomas wrote that "the color-blind Constitution does not bar the government from taking measures to remedy past state-sponsored discrimination – indeed, it requires that such measures be taken in certain circumstances."

Outline

A color-blind society, in sociology, is one in which racial classification does not affect a person's socially created opportunities. A racially color blind society is or would be free from differential legal or social treatment based on race or color. A color-blind society would have race-neutral governmental policies and would reject all racial discrimination.

Racial color blindness reflects a societal ideal that skin color is insignificant. The ideal was most articulated "along with the emergence of the Civil Rights Movement in the US and anti-racist movements abroad". Color-blind ideology is based on tenets of non-discrimination, due process of law, equal protection under the law, and equal opportunities regardless of race, ideas which have strongly influenced Western liberalism in the post-World War II period.

Proponents of "color-blind" practices largely believe that treating people equally as individuals leads to a more equal society or that racism and race privilege no longer exercise the power they once did, rendering the need for policies such as race-based affirmative action obsolete.

Support

Professor William Julius Wilson of Harvard University has argued that "class was becoming more important than race" in determining life prospects within the black community. Wilson has published several works including The Declining Significance of Race (1978) and The Truly Dis-advantaged (1987) explaining his views on black poverty and racial inequality. He believes that affirmative action primarily benefits the most privileged individuals within the black community. This is because strictly race-based programs disregard a candidate's socioeconomic background and therefore fail to help the poorer portion of the black community that actually needs the assistance. He claims that in a society where millions of black people live in the middle and upper classes and millions of white people live in poverty, race is no longer an accurate indication of privilege. Recognizing someone's social class is more important than recognizing someone's race, indicating that society should be class-conscious, not race-conscious, Wilson argues.

In his famous 1963 speech "I Have a Dream", Martin Luther King Jr. proclaimed, "I have a dream that my four little children will one day live in a nation where they will not be judged by the color of their skin but by the content of their character." This statement was widely interpreted as an endorsement of color-blind racial ideology. Roger Clegg, the President of the Center for Equal Opportunity, felt that this quotation supported the idea that race-conscious and equal opportunity should not exist, as he believes people should not be treated differently based on the color of their skin. However, not all agreed with this interpretation. American author Michael Eric Dyson felt that Dr. King only believed in the possibility of a color-blind society under the condition that racism and oppression were ultimately destroyed.

Supreme Court Justice Clarence Thomas has supported color-blind policies. He believes the Equal Protection Clause of the Fourteenth Amendment forbids consideration of race, such as race-based affirmative action or preferential treatment. He believes that race-oriented programs create "a cult of victimization" and imply black people require "special treatment in order to succeed".

When defending new voting rights bills in 2020, Republican Texas legislators claimed that since the process they wanted to establish for voter registration did not involve different processes for people of different races and did not involve collecting information about race or ethnicity, their new requirements for eligibility to vote were "color blind" and should not be considered racially discriminatory.

Some argue that the existence of majority-majority and majority-minority areas are not the result of racial discrimination and that this viewpoint ignores the possibility of other factors underlying residential segregation such as the attitude of realtors, bankers, and sellers.

While the field of whiteness studies often discusses alleged failures of racial color blindness, it has been criticized for its focus on reprimanding the white population, whereas similar fields such as black studies, women's studies, and Chicano studies celebrate the contributions of the eponymous group.

Among conservative presidents, color blindness as an idea has increased in the late 20th century as well as in the 21st century.

Where racial disparities were once explained in terms of biology, they are now being discussed in terms of culture. "Culture" in this framework is seen as something fixed and hard to change. One example form of rhetoric used in this framework is the argument, "if Irish, Jews (or other ethnic groups) have 'made it', how come black people have not?"

Some supporters of racial color blindness argue racial inequality can be supported by relying on cultural, rather than biological, explanations such as "this race has too many babies". Some no longer view racism as a problem under this belief and see government programs targeting race as no longer necessary due to the avoidance of racism. Bonilla-Silva describes naturalization as a frame that portrays racism as a natural outcome of individuals' choices, and "just the way things are". While Bonna-Silva himself disagrees with these as "minimization of racism", these are views common among supporters of racial color blindness.

In response to the global Black Lives Matter movement, the phrase All Lives Matter came into being as a term for racial color blindness. Several notable individuals have supported All Lives Matter, such as NFL cornerback Richard Sherman who said, "I stand by what I said that All Lives Matter and that we are human beings." A 2015 telephone poll in the US found that 78% of respondents said that "all lives matter" was closest to their own personal views. Despite this, the term was criticized by professor David Theo Goldberg as reflecting a view of "racial dismissal, ignoring, and denial."

Criticism

In 1997, Leslie G. Carr published Color-Blind Racism which reviewed the history of racist ideologies in America. He saw "color-blindness" as an ideology that undercuts the legal and political foundation of racial integration and affirmative action.

Stephanie M. Wildman's Privilege Revealed: How Invisible Preference Undermines America, writes that advocates of a meritocratic, race-free worldview do not acknowledge the systems of privilege which benefit them, such as social and financial inheritance. She argues that this inheritance privileges "whiteness", "maleness", and heterosexuality while disadvantaging descendants of slaves.

Sociologist Eduardo Bonilla-Silva writes that majority groups use color-blindness to avoid discussing racism and discrimination. Color-blindness can be seen as a way to undermine minority hardships, as it used to argue that the United States is a meritocracy, in which people succeed only because they work hard and not their privilege. John R. Logan has disagreed with this notion of meritocracy, as the average black or Hispanic household earning more than $75,000 still live in a less affluent neighborhood than a white household earning less than $40,000 and poverty rates are higher for minorities.

Amy Ansell of Bard College argues that color-blindness operates under the assumption that we are living in a world that is "post-race", where race no longer matters. She argues this is not true and if it was that race would not be taken into consideration even when trying to address inequality or remedy past wrongs.

Abstract liberalism utilizes themes from political and economic liberalism, such as meritocracy and the free market, to argue against the strong presence of racism. Some suggest it results in people being for equality in principle but against government action to implement equality, described by some sociologists as laissez-faire racism.

Robert D. Reason and Nancy J. Evans outline a similar description of color-blindness, which is based on four beliefs: 1. Privilege is based on merit. 2. Most do not care about a person's race. 3. Social inequality is due to "cultural deficits" of individual people. 4. Given the previous three beliefs, there is no need to pay "systematic attention" to any current inequities. They argue the prevalence of color-blindness is attributed to lack of knowledge or lack of exposure. They argue that due to racial separation in housing and education many Americans lack direct contact with present racism.

In Social Inequality and Social Stratification in US Society, Christopher Doob argues that racial color blindness's proponents "assert...that they are living in a world where racial privilege no longer exists, but their behavior 'supports' racialized structures and practices".

Eduardo Bonilla-Silva has argued racial color blindness is insufficient to address racial inequality. He argues it involves egalitarianism while opposing concrete proposals to reduce inequality. He has argued it ignores the under-representation of minorities in prestigious institutions, along with institutional practices that encourage segregation.

Eberhardt, Davies, Purdie-Vaughns, and Johnson studied implicit racial biases, suggesting people react differently to faces of members of their race compared to members of other races. They found a correlation between race and judicial outcomes and suggest a color blind approach may not actually be possible.

Research

Fryer et al. argued that color-blind affirmative action is about as efficient as race-conscious affirmative action in the short run but is less profitable in the long term.

In 2010, Apfelbaum et al. exposed elementary school students to color-blind ideology and found that those students were less likely to detect or report overt racial discrimination. The authors argued racial color blindness allows overt racism to persist."

Amy Ansell, a sociologist at Bard College, has compared and contrasted the development of the color-blindness in the United States and South Africa. Given that white people are a minority population in South Africa and a majority population in the United States, Ansell expected to see a significant difference in the manifestation of color-blindness in both countries. The thirty-year time difference between the departure from Jim Crow and cessation of apartheid and differences in racial stratification and levels of poverty also led Ansell to expect a clear difference between the colorblindness ideology in the United States and South Africa. However, she concludes contemporary color-blindness in the two countries is nearly identical.

Vorauer, Gagnon, and Sasaki examined the effect that messages with a color-blind ideology had on white Canadians entering one-on-one interactions with Aboriginal Canadians. White Canadians who heard messages emphasizing color-blind ideology were much more likely to be concerned with ensuring the subsequent interaction did not go badly and were more likely hostile, uncomfortable, and uncertain. White participants who heard messages emphasizing multicultural ideology, or the valuing of people's differences, asked more positive questions focused on the other person more relaxedly.

Alternatives

Researchers also offer alternatives to the color-blindness discourse. Reason and Evans call for people to become "racially cognizant" and continuously acknowledge the role that race plays in their lives. They argue it is important to balance personal identity and a person's race.

Researcher Jennifer Simpson argued that "in setting aside color blindness, White [people] must learn to see, accept...the possibility that some of the good, ease, or rewards they have experienced have not been solely the result of hard work" but from "a system biased in their favor." This conscious exploration of whiteness as a racial and social identity and the acknowledgment of the role of whiteness is connected to modern whiteness studies.

In a recent publication of the academic journal Communication Theory, Jennifer Simpson proposed a "more productive dialogue about race". New dialogue must take a more complex look at race, openly looking at different perspectives on race. Simpson argues white people must engage with other races in discussing the ongoing effects of racism, requiring white people to participate in "communicative behavior that may threaten simultaneously their sense of self and their material power in the social order".

In education

A multisite case study of Atlantic State University, a primarily white institution, and Mid-Atlantic State University, a historically black college, explored color-blind ideologies among the institutions’ white faculty members at the undergraduate and graduate level. In interviews with white faculty members at both institutions, researchers found the faculty often engaged with students from a color-blind perspective, avoiding racial terms but implying them allowed white faculty to label minority students "as academically inferior, less prepared, and less interested in pursuing research and graduate studies while potentially ignoring structural causes" of inequity. The study concludes that color-blind ideology held by school faculty can reduce a student of color's perception of their academic abilities and potential to achieve success in STEM disciplines and in graduate school.

A case study of a suburban, mixed-race high school examined the trend toward color-blind ideology in schools among white faculty. It argued white schoolteachers's color-blind ideology often masks their fears of being accused of racism and prevents a deeper examination of race.

Case studies of three large school districts, (Boston, Massachusetts; Wake County, North Carolina; and Jefferson County, Louisville) found that the districts’ race-neutral, or color-blind, policies to combat school segregation may disadvantage minorities and "reframe privilege as common sense" while ignoring structural inequalities.

Mathematical fallacy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Mathematical_fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.

For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules.

The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry, the five colour theorem of graph theory). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.

Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.

Howlers

${\displaystyle \begin{array}{l}{\displaystyle \frac{d}{dx}}{\displaystyle \frac{1}{x}}\\ ={\displaystyle \frac{d}{d}}{\displaystyle \frac{1}{x^2}}\\ ={\displaystyle \frac{d\mathrm{\setminus }}{d\mathrm{\setminus }}}{\displaystyle \frac{1}{x^2}}\\ =-{\displaystyle \frac{1}{x^2}}\end{array}}$

Anomalous cancellation in calculus

Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a howler. The following is an example of a howler involving anomalous cancellation:

$${\displaystyle \frac{16}{64}=\frac{16/}{6/4}=\frac{1}{4}.}$$

Here, although the conclusion 16/64 = 1/4 is correct, there is a fallacious, invalid cancellation in the middle step. Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial by the matrix.

Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Maxwell. Outside the field of mathematics the term howler has various meanings, generally less specific.

Division by zero

The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number.

1. Let a and b be equal, nonzero quantities

   ${\displaystyle a=b}$

2. Multiply by a

   ${\displaystyle a^2=ab}$

3. Subtract b²

   ${\displaystyle a^2-b^2=ab-b^2}$

4. Factor both sides: the left factors as a difference of squares, the right is factored by extracting b from both terms

   ${\displaystyle (a-b)(a+b)=b(a-b)}$

5. Divide out (a − b)

   ${\displaystyle a+b=b}$

6. Use the fact that a = b

   ${\displaystyle b+b=b}$

7. Combine like terms on the left

   ${\displaystyle 2b=b}$

8. Divide by the non-zero b

   ${\displaystyle 2=1}$

Q.E.D.

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a = b. Since division by zero is undefined, the argument is invalid.

Analysis

Mathematical analysis as the mathematical study of change and limits can lead to mathematical fallacies — if the properties of integrals and differentials are ignored. For instance, a naive use of integration by parts can be used to give a false proof that 0 = 1. Letting u = 1/log x and dv = dx/x, we may write:

${\displaystyle \int \frac{1}{x\log x}dx=1+\int \frac{1}{x\log x}dx}$

after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits a and b.

${\displaystyle {\int }_a^b\frac{1}{x\log x}dx=1{|}_a^b+{\int }_a^b\frac{1}{x\log x}dx=0+{\int }_a^b\frac{1}{x\log x}dx={\int }_a^b\frac{1}{x\log x}dx}$

Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

Multivalued functions

Main article: Multivalued function

Many functions do not have a unique inverse. For instance, while squaring a number gives a unique value, there are two possible square roots of a positive number. The square root is multivalued. One value can be chosen by convention as the principal value; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for nth roots.

Positive and negative roots

Care must be taken when taking the square root of both sides of an equality. Failing to do so results in a "proof" of 5 = 4.

Proof:

Start from

${\displaystyle -20=-20}$

Write this as

${\displaystyle 25-45=16-36}$

Rewrite as

${\displaystyle 5^2-5\times 9=4^2-4\times 9}$

Add 81/4 on both sides:

${\displaystyle 5^2-5\times 9+\frac{81}{4}=4^2-4\times 9+\frac{81}{4}}$

These are perfect squares:

${\displaystyle {\left(5-\frac{9}{2}\right)}^2={\left(4-\frac{9}{2}\right)}^2}$

Take the square root of both sides:

${\displaystyle 5-\frac{9}{2}=4-\frac{9}{2}}$

Add 9/2 on both sides:

${\displaystyle 5=4}$

Q.E.D.

The fallacy is in the second to last line, where the square root of both sides is taken: a² = b² only implies a = b if a and b have the same sign, which is not the case here. In this case, it implies that a = –b, so the equation should read

${\displaystyle 5-\frac{9}{2}=-\left(4-\frac{9}{2}\right)}$

which, by adding 9/2 on both sides, correctly reduces to 5 = 5.

Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity

${\displaystyle {\cos }^{2}x=1-{\sin }^{2}x}$

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

${\displaystyle \cos x=\sqrt{1-{\sin }^{2}x}}$

Evaluating this when x = π , we get that

${\displaystyle -1=\sqrt{1-0}}$

or

${\displaystyle -1=1}$

which is incorrect.

The error in each of these examples fundamentally lies in the fact that any equation of the form

${\displaystyle x^2=a^2}$

where ${\displaystyle a\ne 0}$, has two solutions:

${\displaystyle x=\pm a}$

and it is essential to check which of these solutions is relevant to the problem at hand. In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive. In particular, when x is set to π, the second equation is rendered invalid.

Square roots of negative numbers

Invalid proofs utilizing powers and roots are often of the following kind:

${\displaystyle 1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1.}$

The fallacy is that the rule ${\displaystyle \sqrt{xy}=\sqrt{x}\sqrt{y}}$ is generally valid only if at least one of ${\displaystyle x}$ and ${\displaystyle y}$ is non-negative (when dealing with real numbers), which is not the case here.

Alternatively, imaginary roots are obfuscated in the following:

${\displaystyle i=\sqrt{-1}={\left(-1\right)}^{\frac{2}{4}}={\left({\left(-1\right)}^2\right)}^{\frac{1}{4}}=1^{\frac{1}{4}}=1}$

The error here lies in the third equality, as the rule ${\displaystyle a^{bc}=(a^b{)}^c}$ only holds for positive real a and real b, c.

Complex exponents

When a number is raised to a complex power, the result is not uniquely defined (see Exponentiation § Failure of power and logarithm identities). If this property is not recognized, then errors such as the following can result:

${\displaystyle \begin{array}{rl}{e}^{2\pi i}& =1\\ {\left(e^{2\pi i}\right)}^i& =1^i\\ e^{-2\pi }& =1\end{array}}$

The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being {e^2πn | n ∈ ℤ}.

Geometry

Many mathematical fallacies in geometry arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so.

In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.

Fallacy of the isosceles triangle

The fallacy of the isosceles triangle, from (Maxwell 1959, Chapter II, § 1), purports to show that every triangle is isosceles, meaning that two sides of the triangle are congruent. This fallacy was known to Lewis Carroll and may have been discovered by him. It was published in 1899.

Given a triangle △ABC, prove that AB = AC:

1. Draw a line bisecting ∠A.
2. Draw the perpendicular bisector of segment BC, which bisects BC at a point D.
3. Let these two lines meet at a point O.
4. Draw line OR perpendicular to AB, line OQ perpendicular to AC.
5. Draw lines OB and OC.
6. By AAS, △RAO ≅ △QAO (∠ORA = ∠OQA = 90°; ∠RAO = ∠QAO; AO = AO (common side)).
7. By RHS, △ROB ≅ △QOC (∠BRO = ∠CQO = 90°; BO = OC (hypotenuse); RO = OQ (leg)).
8. Thus, AR = AQ, RB = QC, and AB = AR + RB = AQ + QC = AC.

Q.E.D.

As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.

The error in the proof is the assumption in the diagram that the point O is inside the triangle. In fact, O always lies on the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide). Furthermore, it can be shown that, if AB is longer than AC, then R will lie within AB, while Q will lie outside of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts). Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.

Proof by induction

There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing that if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this, it can be shown to be true for all cases. The following "proof" shows that all horses are the same colour.

1. Let us say that any group of N horses is all of the same colour.
2. If we remove a horse from the group, we have a group of N − 1 horses of the same colour. If we add another horse, we have another group of N horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of N horses.
3. Thus we have constructed two groups of N horses all of the same colour, with N − 1 horses in common. Since these two groups have some horses in common, the two groups must be of the same colour as each other.
4. Therefore, combining all the horses used, we have a group of N + 1 horses of the same colour.
5. Thus if any N horses are all the same colour, any N + 1 horses are the same colour.
6. This is clearly true for N = 1 (i.e., one horse is a group where all the horses are the same colour). Thus, by induction, N horses are the same colour for any positive integer N, and so all horses are the same colour.

The fallacy in this proof arises in line 3. For N = 1, the two groups of horses have N − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of N + 1 = 2 horses is not necessarily all of the same colour. The implication "every N horses are of the same colour, then N + 1 horses are of the same colour" works for any N > 1, but fails to be true when N = 1. The basis case is correct, but the induction step has a fundamental flaw.