Modulus, also known as absolute value, is a mathematical operation that determines the magnitude or distance of a number from zero. It is denoted by two vertical bars surrounding the number. The modulus of a number is always positive or zero, regardless of the sign of the original number.
The concept of modulus has been present in mathematics for centuries. The ancient Greeks and Egyptians used the concept of absolute value in their calculations, although it was not formally defined as modulus until later. The term "modulus" was first introduced by the German mathematician Carl Friedrich Gauss in the early 19th century.
Modulus is typically introduced in middle school or early high school mathematics, around grades 7-9. It is an important concept in algebra and number theory.
Modulus involves the following knowledge points:
Absolute value: The absolute value of a number is its distance from zero on the number line. It is always positive or zero.
Sign: The sign of a number indicates whether it is positive, negative, or zero.
To calculate the modulus of a number, follow these steps:
Identify the number for which you want to find the modulus.
If the number is positive or zero, the modulus is equal to the number itself.
If the number is negative, ignore the negative sign and the modulus is equal to the positive value of the number.
For example, the modulus of -5 is 5, and the modulus of 8 is 8.
There are no specific types of modulus. However, modulus can be applied to various mathematical operations, such as addition, subtraction, multiplication, and division.
The properties of modulus include:
Non-negativity: The modulus of a number is always non-negative, meaning it is either positive or zero.
Symmetry: The modulus of a number is symmetric, meaning the modulus of -x is the same as the modulus of x.
Triangle inequality: The modulus of the sum of two numbers is less than or equal to the sum of their individual moduli. In other words, |a + b| ≤ |a| + |b|.
To find or calculate the modulus of a number, simply remove the negative sign (if present) and consider the absolute value of the number.
For example, the modulus of -7 is 7, and the modulus of 0 is 0.
The formula for modulus can be expressed as:
|a| = a, if a ≥ 0 |a| = -a, if a < 0
To apply the modulus formula, substitute the given number into the formula and follow the rules for positive and negative values.
For example, to find the modulus of -9, apply the formula as follows:
| -9 | = -(-9) = 9
The symbol or abbreviation for modulus is two vertical bars surrounding the number, as in |a|.
The methods for modulus include:
Direct calculation: Simply remove the negative sign (if present) and consider the absolute value of the number.
Using the modulus formula: Apply the formula |a| = a, if a ≥ 0 or |a| = -a, if a < 0.
Example 1: Find the modulus of -12. Solution: | -12 | = -(-12) = 12
Example 2: Calculate the modulus of 0. Solution: | 0 | = 0
Example 3: Determine the modulus of 5. Solution: | 5 | = 5
Question: What is modulus? Answer: Modulus, also known as absolute value, is a mathematical operation that determines the magnitude or distance of a number from zero. It is always positive or zero, regardless of the sign of the original number.