Problem 40 - Introductory Algebra for College Students

Question

find the multiplicative inverse of each $$-12$$

Step-by-Step Solution

Answer

The multiplicative inverse of \(-12\) is \(-1/12\).

Step 1: Understanding Multiplicative Inverse

The multiplicative inverse of a number is another number such that when both numbers are multiplied, the product is \(1\). It is often denoted as \(1/x\) or \(x^{-1}\), where \(x\) is the original number.

Step 2: Find the Multiplicative Inverse of -12

The multiplicative inverse of a number \(x\) is \(1/x\). By applying this principle, the multiplicative inverse of \(-12\) would be \(-1/12\) because \(-12\) times \(-1/12\) equals \(1\).

Key Concepts

Inverse Operations

Inverse operations are pairs of mathematical actions that undo each other. Multiplication and division are a classic example:

• Multiplication involves increasing a number by a certain factor.
• Division, on the other hand, breaks that factor down, reverting to the original number.

A practical use of inverse operations is in finding the multiplicative inverse, also known as the reciprocal, of a number. For any non-zero number like \(-12\), its multiplicative inverse is \(-1/12\), since multiplying these two gives \(1\). This property helps solve equations and can simplify fractions.

Negative Numbers

Negative numbers extend the number line below zero. They are just as important in mathematics as positive numbers:

• Opposite in operation, a negative number multiplied by another negative number returns a positive product.
• However, if multiplied by a positive, the result is negative.

In terms of multiplicative inverse, \(-12\) behaves like any other number. Its inverse is \(-1/12\), because two negatives multiplied yield a positive, making their product \(1\). Handling negatives accurately is fundamental as they occur in various mathematical operations and real-world contexts.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(beq0\). They include:

• Integers like \(-12\), which can be written as \(\frac{-12}{1}\).
• And non-integers like \(-1/12\).

Finding the multiplicative inverse involves flipping the fraction. For our number \(-12\), its structure as \(\frac{-12}{1}\) makes its inverse \(\frac{-1}{12}\). Working with rational numbers helps recognize their versatility, as they encompass negative values and their inverses, aiding in clearer mathematical solutions.