inequality

Physics

A relationship between two quantities in which one of the quantities is not equal to (or not necessarily equal to) the other quantity. If the quantities are a and b, two inequalities exist: a is greater than b, written a > b, and a is less than b, i.e. a < b. Similar statements can take the form: a is greater than or equal to b, written a ≥ b, and a is less than or equal to b, which is denoted a ≤ b. There are many applications of inequalities in physical science, an example being the Heisenberg uncertainty principle.

Mathematics

The following symbols have the meanings shown:
- ≠ is not equal to,
- < is less than,
- ≤ is less than or equal to,
- > is greater than,
- ≥ is greater than or equal to.
An inequality is a statement of one of the forms: a ≠ b, a<b, a ≤ b, a>b or a≥b, where a and b are suitably comparable expressions. Inequalities involving<and > are called ‘strict’, and those involving≤and≥are termed ‘weak’.

Given an inequality involving one real variable x, the problem may be to find the solutions, that is, the values of x that satisfy the inequality, in an explicit form. Often, the set of solutions, known as the solution set, may be given as an interval or as a union of intervals. For example, the solution set of the inequality

$x - 2 > \frac{6}{x + 3}$

is the set (−4, −3) ⋃ (3, ∞).

(Some authors use ‘inequation’ for a statement which holds only for some values of the variables involved. For them, an inequation has a solution set but an inequality does not; an inequality is comparable to an identity such as x² + 2 > 2x which holds for all real x)

Astronomy

A variation in the movement of a celestial object in its orbit about another which cannot be accounted for by their mutual gravitational attraction. Inequalities usually arise because of the perturbing forces of one or more other massive objects in the system. For example, the so-called great inequality in the orbital movements of the planets Jupiter and Saturn about the Sun is an oscillation in their heliocentric longitude with a period of some 900 years caused by their mutual perturbations and also by the nearly 2 : 5 resonance in their mean motions. See also Lunar Inequality.

Computer

A binary relation that typically expresses the relative magnitude of two quantities, usually numbers though more generally elements of a partially ordered set (see partial ordering).
The inequalities defined on the integers usually include
- < (less than)
- ≤ (less than or equal to)
- > (greater than)
- ≥ (greater than or equal to)
- ≠ (not equal to)
A similar set of inequalities is usually defined on the real numbers; such inequalities can produce errors when used in programming languages because of the inherent inaccuracies in the way real numbers are usually represented.
The term inequality is often applied to any comparison involving algebraic expressions and using the above symbols. A special case is the triangle inequality:
$| a + b | \leq | a | + | b |$
where | | denotes the absolute value function.

Geography

The fact of global inequality is not disputed; the world is routinely divided into developed/developing, North/South, more/less/least economically developed, high/medium/low development, and so on. The richest 2% of adults in the world own more than half of global household wealth (J. B. Davies, S. Sandstrom, and A. Shorrocks 2006). There is no consensus about the causes of global inequality, not least because of the difficulties in measuring inequality: if we scale income by some index of health, there is more inequality in the world than if we consider income alone (Deaton (2006) NBER W. Paper W12735).

Many writers blame globalization: Ash (2004, TIBG 29, 2) argues that ‘TNCs and international banks are deeply embedded in the production of inequality, through their influence over governments, their tax evasion schemes, their poor environmental record, their labour market practices, their locational mobility, their market power, their dumping practices.’ Cornia and Court (2001, Policy Brief 4 World Inst. Dev. Econ. Res.) contend that the liberalization of domestic banking associated with globalization has caused rises in income inequality; Rowson (2000) healthmatters 41 agrees.

There is also debate over the impact of structural adjustment measures on inequality. In 2004, SAPRIN claimed that structural adjustment polices are the largest single cause of inequality; at the other end of the spectrum, Ferreira (2004) World Bank Policy Working Paper 1641 asserts that, under a programme of structural adjustment, the population of Tanzania in poverty fell from 65% in 1983 to 51% in 1991.

Whether global inequality is increasing or decreasing is by no means clear. The UNDP (2005) reports that the ratio of the assets of the world’s richest 20% to the poorest 20% increased from 30 : 1 in 1960 to 78 : 1 in 1994, yet Firebaugh and Goesling (2004) Amer. Journal Soc. 110, 2 report that global income inequality declined in the last decades of the 20th century.

There are those who hold that there is a relationship between regional inequality (as measured by the Gini coefficient) and level of development (as measured by per capita GDP), but the data in the table do not offer robust support to this view.

Region	Gini coefficient for per capita GDP (ppp US$) 2003	GDP per capita (ppp US$) 2003
Sub-Saharan Africa	72.2	1 856
South Asia	33.4	2 897
East Asia and Pacific	52.0	5 100
Latin America and Caribbean	57.1	7 404
High Income OECD	67.0	30 181

Source: UNDP (2005) 55 and 222.

National inequality
has grown markedly; since 1980, virtually every country in the world has become more unequal (Gilbert (2007) Geog. Compass 1, 3). During the 1980s and 1990s, income inequality rose in the OECD countries, particularly escalated in China, Vietnam, and South Asia, shot up in Latin America, and went ballistic in Russia in the 1990s (2005, World Bank 45). See Alderson and Nielsen (2002) Amer. Journal Soc. 107 for an exhaustive study of inequality in sixteen OECD countries.
In Britain, inequality rose in the 1980s and 1990s. Changes since 2000 are less clear, but the proportions of ‘exclusive wealthy’—the people with so much wealth that they can exclude themselves from the norms of society—are at their highest since 1990 (Joseph Rowntree Foundation 2007). The rich and poor now live further apart; while in some cities over half of all households are now breadline poor, wealthy households have concentrated in the outskirts and surrounds of major cities, and exclusive wealthy households have increasingly been gathering around London. Since 2000, there seems to have been little progress in reducing geographical polarization.
Does inequality matter? Gilbert (2007) Geog. Comp. 1, 3 argues that inequality creates social ghettos and increased violence, and blocks progress for the poor. ‘Inequality is not inherently wrong—as long as three conditions are met: society as a whole is getting richer; there is a safety net for the very poor; everybody, regardless of class, race, creed, or sex, has an opportunity to climb up through the system’ (The Economist, 15 June 2006). Korpi (2008) J. Econ. Geog. 8, 2 finds a positive link between wage inequality and labour market size, due to increasing upper income levels as labour market size increases. See D. Dorling (2011).

www.jrf.org.uk/knowledge/findings/housing/2077.asp The Joseph Rowntree Foundation study of poverty and wealth across Britain.

Philosophy

See equality.

Economics

Differences in the distribution of economic stocks or flows among economic agents. For example, wealth inequality refers to the distribution of the stock of wealth, whereas income inequality refers to the distribution of the flow of income. Inequality can arise among individuals or groups within an economy, or among nations. Inequality can be graphically represented by the Lorenz curve or measured by a range of indicators including the Atkinson index and the Gini coefficient.