absolute value

Examples of absolute value in the following topics:

• Absolute Value

  • Absolute value can be thought of as the distance of a real number from zero.
  • For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.
  • The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English.
  • Other names for absolute value include "numerical value," "modulus," and "magnitude."
  • The absolute values of 5 and -5 shown on a number line.

• Equations with Absolute Value

  • To solve an equation with an absolute value, first isolate the absolute value, and then solve for the positive and negative cases.
  • At face value, nothing could be simpler: absolute value simply means the distance a number is from zero.  
  • The absolute value of $-5$ is $5$, and the absolute value of $5$ is also $5$, since both $-5$ and $5$ are $5$ units away from $0$.
  • Recall that absolute value is a measure of distance, so it can never be a negative value.
  • The following steps describe how to solve an absolute value equation:

• Inequalities with Absolute Value

  • Inequalities with absolute values can be solved by thinking about absolute value as a number's distance from 0 on the number line.
  • More complicated absolute value problems should be approached in the same way as equations with absolute values: algebraically isolate the absolute value, and then algebraically solve for $x$.
  • It is difficult to immediately visualize the meaning of this absolute value, let alone the value of $x$ itself.
  • Now think: the absolute value of the expression is greater than –3.
  • Absolute values are always positive, so the absolute value of anything is greater than –3!

• Absolute Convergence and Ratio and Root Tests

  • An infinite series of numbers is said to converge absolutely if the sum of the absolute value of the summand is finite.
  • An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite.
  • Similarly, an improper integral of a function, $\textstyle\int_0^\infty f(x)\,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\int_0^\infty \left|f(x)\right|dx = L$.
  • (A convergent series that is not absolutely convergent is called conditionally convergent.)
  • State the conditions when an infinite series of numbers converge absolutely

• The Third Law

  • According to the third law of thermodynamics, the entropy of a perfect crystal at absolute zero is exactly equal to zero.
  • The third law of thermodynamics is sometimes stated as follows: The entropy of a perfect crystal at absolute zero is exactly equal to zero.
  • Nernst proposed that the entropy of a system at absolute zero would be a well-defined constant.
  • This law provides an absolute reference point for the determination of entropy. ( diagrams the temperature entropy of nitrogen. ) The entropy (S) determined relative to this point is the absolute entropy represented as follows:
  • Absolute value of entropy can be determined shown here, thanks to the third law of thermodynamics.

• Absolute Zero

  • Absolute zero is the coldest possible temperature; formally, it is the temperature at which entropy reaches its minimum value.
  • Absolute zerois the coldest possible temperature.
  • Formally, it is the temperature at which entropy reaches its minimum value.
  • Absolute zero is universal in the sense that all matteris in ground state at this temperature .
  • Explain why absolute zero is a natural choice as the null point for a temperature unit system

• Median and Mean

  • State whether it is the mean or median that minimizes the mean absolute
  • In the section "What is central tendency," we saw that the center of a distribution could be defined three ways: (1) the point on which a distribution would balance, (2) the value whose average absolute deviation from all the other values is minimized, and (3) the value whose squared difference from all the other values is minimized.
  • The mean is the point on which a distribution would balance, the median is the value that minimizes the sum of absolute deviations, and the mean is the value that minimizes the sum of the squared deviations.
  • You can see that the sum of absolute deviations from the median (20) is smaller than the sum of absolute deviations from the mean (22.8).
  • Absolute and squared deviations from the median of 4 and the mean of 6.8

• The Third Law of Thermodynamics and Absolute Energy

  • The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero.
  • The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches zero.
  • At absolute zero there is only 1 microstate possible (Ω=1) and ln(1) = 0.
  • The constant value (not necessarily zero) is called the residual entropy of the system.
  • The entropy determined relative to this point (absolute zero) is the absolute entropy.

• Gauge Pressure and Atmospheric Pressure

  • Pressure is often measured as gauge pressure, which is defined as the absolute pressure minus the atmospheric pressure.
  • The situation changes when extreme vacuum pressures are measured; absolute pressures are typically used instead.
  • To find the absolute pressure of a system, the atmospheric pressure must then be added to the gauge pressure.
  • While gauge pressure is very useful in practical pressure measurements, most calculations involving pressure, such as the ideal gas law, require pressure values in terms of absolute pressures and thus require gauge pressures to be converted to absolute pressures.
  • Explain the relationship among absolute pressure, gauge pressure, and atmospheric pressure

• Absolute Advantage and the Balance of Trade

  • Absolute advantage and balance of trade are two important aspects of international trade that affect countries and organizations.
  • Absolute advantage and balance of trade are two important aspects of international trade that affect countries and organizations .
  • Adam Smith first described the principle of absolute advantage in the context of international trade, using labor as the only input.
  • Since absolute advantage is determined by a simple comparison of labor productivities, it is possible for a party to have no absolute advantage in anything; in that case, according to the theory of absolute advantage, no trade will occur with the other party.
  • The balance of trade (or net exports, sometimes symbolized as NX) is the difference between the monetary value of exports and imports in an economy over a certain period.