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Absolute value refers to a number’s distance from zero on the number line, without taking direction into account. A number’s absolute value can never be negative. The absolute value of 5 is 5, for example, and the absolute value of 5 is also 5. The absolute value of a number can be conceived of as its position on the real number line in relation to zero. Furthermore, the distance between two real numbers is the absolute value of their difference. After discussing the absolute value definition, let’s discuss its history!
History
The concept of absolute value dates back to 1806 when Jean-Robert Argand coined the term module to represent the complicated absolute value. In 1857, the English spelling was changed to modulus. In 1841, Karl Weierstrass invented the vertical bar notation. Although the term modulus is still used occasionally, absolute value and magnitude are interchangeable.
Absolute Value Symbol
We write a vertical bar on either side of a number (or a variable) to denote its absolute value, i.e., |x|, where x is an integer. The absolute value of 4 is expressed as |4|, for example. In addition, the absolute value of -4 is denoted by the symbol |-4|. As previously stated, the absolute value always yields a non-negative value. As a result, |4|=|-4| =4. That is, it converts negative integers to positive numbers as well.
Absolute Value for Complex Numbers
Although the modulus idea was originally applied to complex numbers, students are taught about absolute value in the context of real numbers. Using the Pythagorean theorem, the absolute value of a complex integer is defined as its distance from the origin on a complex plane.
The square root of x2 + y2 is the absolute value of z for any complex integer, where x is a real number and y is an imaginary number:
|z| is equal to (x2 + y2)1/2
When the imaginary component of the number is zero, the definition corresponds to the typical description of a real number’s absolute value.
Properties of the Absolute Value
Multiplicativity, Non-negativity, subadditivity, and positive-definiteness are the four fundamental qualities of absolute value. While these features may appear to be difficult, they are simple to comprehend when using examples.
|a| ≥ 0: non-negativity refers to a number’s absolute value being larger than or = to zero.
|a| = 0 ⇔ a = 0: Positive-definiteness denotes that a number’s absolute value is zero only if the number is zero.
|ab| = |a| |b|: The absolute value of a product of two numbers equals the product of the absolute values of each number, which is known as multiplication. For example, |(2)(-3)| = |2| |-3| =(2)(3) = 6
|a + b| ≤ |a| + |b|: The absolute value of the sum of two real numbers is less than or equal to the total of the absolute values of the two numbers, according to subadditivity. For example, |2 + -3| ≤ |2| + |-3| because 1 ≤ 5.
The identity of the indiscernible, Idempotence, symmetry, triangle inequality and division preservation are all significant features.
||a|| = |a|: The absolute value of the absolute value is the absolute value, according to idempotence.
|-a| = |a|: The absolute value of a negative number is the same as the absolute value of its positive value, according to symmetry.
|a – b| = 0 ⇔ a = b: Positive-definiteness is an analogous phrase for identity of indiscernible. When both a and b have the same value, the absolute value of a – b is zero.
|a – b| ≤ |a – c| + |c – b|: Subadditivity is equivalent to the triangle of inequality.
|a / b| = |a| / |b| if b ≠ 0: Multiplicativeness is the same as division preservation.
How to Solve Absolute Value Equations?
Absolute value equations are simple to solve. Keep in mind that the absolute value of a positive and negative integer can be the same. To build correct expressions, use the attributes of the absolute value.
1. Separate the absolute value expression from the rest of the expression.
2. Solve the expression in absolute value notation so that it can be expressed as a positive (+) or negative (-) quantity.
3. Find a solution for the unknown.
4. Double-check your work by graphing it or inserting the answers into the equation.
Conclusion
A number’s absolute value is its distance from zero. Absolute value can also be thought of as the size of a number — without taking into account its sign. Model problems with a vertical or horizontal number line to help you understand the absolute value. The difference between zero and the absolute value of the difference between numbers will be clear with the help of the number line. Absolute value is represented by a vertical line on either side of the amount, whether it is a number or a variable.
