Absolute Value

Definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a fundamental operation that is essential in understanding and working with real numbers, radicals, and sequences.

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 from zero on the number line.
2. Absolute value is denoted by vertical bars surrounding the number, e.g., |x| or |5|.
3. 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.
4. Absolute value is used to solve equations and inequalities, as well as to find the distance between two points on the number line.
5. In the context of radicals, absolute value is used to ensure that the result is always non-negative, even for negative radicands.

Review Questions

• Explain how the concept of absolute value is used in the context of real numbers.
  • The concept of absolute value is essential in the study of real numbers. Absolute value represents the distance of a number from zero on the number line, regardless of the number's sign. This means that 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. Absolute value is used to solve equations and inequalities involving real numbers, as well as to find the distance between two points on the number line.
• Describe the role of absolute value in the context of radicals and rational exponents.
  • In the context of radicals and rational exponents, absolute value is used to ensure that the result is always non-negative. For example, when taking the square root of a negative number, the absolute value is used to ensure that the result is a positive number. This is important because the square root of a negative number is an imaginary number, which is not part of the real number system. Similarly, when working with rational exponents, the absolute value is used to ensure that the result is a real number, even for negative bases.
• Analyze how the concept of absolute value is applied in the study of geometric sequences.
  • Absolute value is an important concept in the study of geometric sequences, as it is used to determine the rate of growth or decay of the sequence. In a geometric sequence, each term is a constant multiple of the previous term, creating a pattern of growth or decay. The absolute value of the common ratio, which is the constant multiplier, determines whether the sequence is growing (common ratio greater than 1) or decaying (common ratio less than 1). Additionally, the absolute value of the common ratio is used to determine the behavior of the sequence, such as whether it converges or diverges.

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.

Radical: A mathematical symbol (√) that indicates the square root or nth root of a number.

Geometric Sequence: A sequence where each term is a constant multiple of the previous term, creating a pattern of growth or decay.