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🔟elementary algebra review

Reciprocals

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

Reciprocals refer to the relationship between two numbers where their product is equal to 1. In other words, if two numbers are reciprocals, multiplying them together will always result in 1. Reciprocals are an important concept in both adding and subtracting fractions, as well as solving equations using the division and multiplication properties of equality.

5 Must Know Facts For Your Next Test

  1. The reciprocal of a number is found by flipping the numerator and denominator of a fraction, or by dividing 1 by the number.
  2. Reciprocals are used to simplify fractions by multiplying the numerator and denominator by the reciprocal of the denominator.
  3. When adding or subtracting fractions, the denominators must be the same. This is often achieved by finding the least common denominator and converting the fractions to equivalent fractions with the same denominator.
  4. Reciprocals are used to solve equations by dividing both sides of the equation by the same number to isolate the variable.
  5. The product of a number and its reciprocal is always 1, making reciprocals the multiplicative inverse of a number.

Review Questions

  • How are reciprocals used to simplify fractions when adding and subtracting?
    • Reciprocals are used to simplify fractions when adding and subtracting by converting the fractions to equivalent fractions with a common denominator. This is done by multiplying the numerator and denominator of each fraction by the reciprocal of the other fraction's denominator. This ensures that the denominators are the same, allowing the fractions to be added or subtracted directly.
  • Explain how the concept of reciprocals is applied when solving equations using the division and multiplication properties of equality.
    • When solving equations using the division and multiplication properties of equality, reciprocals are used to isolate the variable. By dividing both sides of the equation by the same number, the variable can be isolated. This is because the reciprocal of a number, when multiplied by that number, results in the multiplicative identity of 1. This allows the variable to be solved for, as the operations on both sides of the equation cancel out, leaving only the variable.
  • Analyze how the relationship between a number and its reciprocal can be used to demonstrate the multiplicative identity property.
    • The relationship between a number and its reciprocal can be used to demonstrate the multiplicative identity property, which states that the product of any number and 1 is equal to the original number. This is because the reciprocal of a number is defined as 1 divided by that number. When a number is multiplied by its reciprocal, the result is always 1, the multiplicative identity. This property is fundamental in solving equations using the division and multiplication properties of equality, as it allows for the isolation of variables by dividing or multiplying both sides of the equation by the same number or its reciprocal.