Problem 31 - Algebra 2

Question

Identify the additive inverse and multiplicative inverse for each number. $$ -\frac{5}{8} $$

Step-by-Step Solution

Answer

Additive inverse: \(\frac{5}{8}\), Multiplicative inverse: \(-\frac{8}{5}\).

Step 1: Understanding the Additive Inverse

To find the additive inverse of a number, we need to find another number such that when it is added to the original number, the sum is zero. For a number \(-\frac{5}{8}\), the additive inverse would be its opposite sign, which is \(\frac{5}{8}\).

Step 2: Finding the Additive Inverse

The equation for the additive inverse is \(-\frac{5}{8} + x = 0\). Solving for \(x\), we add \(\frac{5}{8}\) to both sides: \(x = \frac{5}{8}\). Thus, the additive inverse of \(-\frac{5}{8}\) is \(\frac{5}{8}\).

Step 3: Understanding the Multiplicative Inverse

The multiplicative inverse of a number \(a\) is another number \(b\) such that \(a \times b = 1\). For fractions, the multiplicative inverse can be found by swapping the numerator and denominator.

Step 4: Finding the Multiplicative Inverse

For \(-\frac{5}{8}\), the multiplicative inverse is the reciprocal. So, \(-\frac{5}{8}\) becomes \(-\frac{8}{5}\). Thus, the multiplicative inverse of \(-\frac{5}{8}\) is \(-\frac{8}{5}\).

Key Concepts

Additive Inverse

In mathematics, the additive inverse of a number is the number that, when added to the original, results in zero. Consider any number, and imagine its position on the number line. To find its additive inverse, simply move the same distance in the opposite direction from zero. This brings you to the number that "cancels out" the original number.
For example, with \(-\frac{5}{8}\), the additive inverse is \(\frac{5}{8}\). Why? Because \(-\frac{5}{8} + \frac{5}{8} = 0\). Here’s a quick guide to finding the additive inverse:

• If the number is positive, like \(\frac{5}{8}\), its additive inverse is negative: \(-\frac{5}{8}\).
• If the number is negative, like \(-\frac{5}{8}\), its additive inverse is positive: \(\frac{5}{8}\).

Finding additive inverses can simplify problem-solving, especially when dealing with equations.

Multiplicative Inverse

The multiplicative inverse is another important concept. It is the number that, when multiplied by the original number, yields a product of one. For example, if you have a number \(a\), its multiplicative inverse is \(b\) such that \(a \times b = 1\).
To find the multiplicative inverse of a fraction, you simply "flip" the fraction. This means you switch the numerator and the denominator.
For \(-\frac{5}{8}\), the multiplicative inverse is \(-\frac{8}{5}\). This is because \(-\frac{5}{8} \times -\frac{8}{5} = 1\).

• Positive fractions, when inverted, remain positive.
• Negative fractions, like \(-\frac{5}{8}\), when inverted stay negative, giving us \(-\frac{8}{5}\).

Knowing how to find multiplicative inverses is especially useful when solving equations that require "undoing" a multiplication.

Fractions

Fractions represent parts of a whole. They are written as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. They tell us how many parts we have (the numerator) and how many those parts make up a whole (the denominator).
Consider \(-\frac{5}{8}\):

• The numerator is \(-5\), indicating we have negative five parts.
• The denominator is \(8\), meaning the whole is divided into eight equal parts.

Fractions can also be negative, which signifies value in the opposite direction on the number line, as with \(-\frac{5}{8}\). Understanding fractions is key to performing arithmetic operations like addition, subtraction, multiplication, and division.

Reciprocal

The reciprocal of a number is essentially its multiplicative inverse. When you hear "reciprocal," think about flipping a fraction. This term is interchangeably used with multiplicative inverse but is often more specific to fractions.
When converting a number to its reciprocal, switch the numerator and the denominator. For \(-\frac{5}{8}\), the reciprocal is \(-\frac{8}{5}\). Both the numerator and the denominator are important here:

• The numerator \(-8\) takes \(-5\)'s place, and vice versa.
• Since we started with a negative fraction, the reciprocal remains negative.

Understanding the reciprocal helps in division of fractions, as dividing by a fraction is the same as multiplying by its reciprocal.