Absolute Value

Definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of whether the number is positive or negative. It is a fundamental operation that is used in various mathematical contexts, including integer operations, real number properties, and more.

5 Must Know Facts For Your Next Test

1. The absolute value of a number is always a non-negative value, as it represents the distance of the number from zero on the number line.
2. Absolute value is denoted by vertical bars surrounding the number, such as |x|, which reads as 'the absolute value of x'.
3. Absolute value is used to determine the magnitude or size of a number, regardless of its sign.
4. Absolute value is an important concept in integer operations, as it helps simplify the addition, subtraction, multiplication, and division of signed numbers.
5. The properties of absolute value, such as |x| = |-x|, are essential in understanding the behavior of real numbers and their operations.

Review Questions

• How does the absolute value of a number relate 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. For example, the absolute value of 5 is 5, as it is 5 units away from zero on the positive side of the number line. Similarly, the absolute value of -3 is also 3, as it is 3 units away from zero on the negative side of the number line. Regardless of the sign, the absolute value always gives the distance from zero.
• Explain how absolute value is used in the context of integer operations, such as addition and subtraction.
  • When adding or subtracting integers, the absolute value can be used to simplify the process. For example, to find the difference between -7 and 4, we can use the absolute value to calculate |(-7) - 4| = |-11| = 11. This is because the absolute value removes the sign, allowing us to focus on the distance between the two numbers on the number line. Similarly, when adding integers with different signs, the absolute value can help determine the correct sign of the result.
• Analyze the role of absolute value in the properties of real numbers and how it contributes to our understanding of these properties.
  • The properties of real numbers, such as the commutative, associative, and distributive properties, are closely tied to the concept of absolute value. For instance, the property |x| = |-x| demonstrates the symmetry of absolute value around the origin on the number line. This property, along with others like |x + y| ≤ |x| + |y|, help us understand the behavior of real numbers and their operations in a more comprehensive way. Absolute value is a fundamental tool that allows us to reason about the properties of real numbers and their relationships.

Related terms

Signed Number: A number that has a sign, either positive or negative, indicating its direction on the number line.

Number Line: A visual representation of the set of real numbers, where positive numbers are to the right of zero and negative numbers are to the left.

Distance: The length of the path between two points, which is always a non-negative value.