Absolute Value (Modulus)

Definition

The absolute value, also known as the modulus, of a real number is the distance of that number from zero on the number line. It represents the magnitude or size of a number, regardless of its sign. The absolute value of a number is always a non-negative real number.

5 Must Know Facts For Your Next Test

1. The absolute value of a number is denoted by vertical bars, such as |x|, and represents the distance of the number from zero on the number line.
2. The absolute value of a positive number is the number itself, while the absolute value of a negative number is the opposite of the number.
3. The absolute value function is an important tool in the complex number system, as it allows for the calculation of the magnitude or modulus of a complex number.
4. Absolute value is used in various mathematical operations, such as solving inequalities, finding distances, and defining the properties of functions.
5. The absolute value function satisfies the triangle inequality, which states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side.

Review Questions

• Explain how the absolute value of a number relates to its position on the number line.
  • The absolute value of a number represents the distance of that number from zero on the number line. This means that the absolute value of a number is always a non-negative value, as it measures the magnitude or size of the number, regardless of its sign. For example, the absolute value of 5 is 5, while the absolute value of -5 is also 5, as both numbers are the same distance from zero on the number line.
• Describe how the absolute value function is used in the complex number system.
  • In the complex number system, the absolute value (or modulus) of a complex number $z = a + bi$ is defined as $|z| = \sqrt{a^2 + b^2}$. This represents the distance of the complex number from the origin (0, 0) on the complex plane. The absolute value of a complex number is a crucial concept in understanding the properties and operations of complex numbers, as it allows for the calculation of the magnitude or size of a complex number.
• Analyze how the absolute value function satisfies the triangle inequality and explain the significance of this property.
  • The absolute value function satisfies the triangle inequality, which states that for any three real numbers $a$, $b$, and $c$, the sum of the absolute values of any two of the numbers must be greater than or equal to the absolute value of the third number. Mathematically, this can be expressed as $|a| + |b| \geq |c|$. This property is significant because it allows for the calculation of distances and the definition of various mathematical functions and inequalities involving absolute values. The triangle inequality is a fundamental concept in geometry and has applications in various areas of mathematics, including complex analysis and vector spaces.

Related terms

Real Numbers:

The set of all numbers that can be represented on the number line, including positive and negative integers, fractions, and irrational numbers.

Number Line:

A visual representation of the set of real numbers, where each number is assigned a unique position along a horizontal line.

Distance:

The length of the straight line segment between two points, which represents the magnitude of the difference between the two numbers.