Problem 382 - Intermediate Algebra

Question

What is the difference between the additive inverse and the multiplicative inverse of a number?

Step-by-Step Solution

Answer

The additive inverse of a number results in zero when added to it, while the multiplicative inverse results in one when multiplied by it.

Step 1: Understand the Additive Inverse

The additive inverse of a number is what you add to the original number to get zero. For a number , its additive inverse is . The formula for this is: - n = 0.

Step 2: Understand the Multiplicative Inverse

The multiplicative inverse (or reciprocal) of a number is what you multiply the original number by to get one. For a number , its multiplicative inverse is . The formula for this is: .

Step 3: Compare Additive and Multiplicative Inverses

The main difference between the additive and multiplicative inverses is in their operations and results. The additive inverse of a number results in zero when added to the original number, while the multiplicative inverse results in one when multiplied by the original number.

Key Concepts

Additive Inverse

The additive inverse of a number is simply the value that, when added to the original number, results in zero. For any number, we can find its additive inverse by changing its sign.
For example:

• If the original number is 5, its additive inverse is -5.
  This is because 5 + (-5) = 0.
• If the original number is -3, its additive inverse is 3.
  This is because -3 + 3 = 0.

The general formula is:
- n = 0.
So, the additive inverse of a number 'n' is simply '-n'.
This concept is crucial in solving equations where you want to eliminate a number by adding its inverse.

Multiplicative Inverse

The multiplicative inverse of a number, also known as the reciprocal, is the value that, when multiplied by the original number, results in one.
For any nonzero number , we can find its multiplicative inverse by taking 1 divided by that number.
For example:

• If the original number is 4, its multiplicative inverse is 1/4.
  This is because 4 × 1/4 = 1.
• If the original number is 1/2, its multiplicative inverse is 2.
  This is because 1/2 × 2 = 1.

The general formula is:
= 1.
Thus, the multiplicative inverse of a number 'n' is '1/n'.
This concept is vital in fraction division and solving algebraic equations where you need to isolate a variable.

Reciprocal

The term reciprocal is another way to describe the multiplicative inverse.
When you hear the word reciprocal, think of 'flipping' the number.
For example, the reciprocal of a fraction like 3/4 is 4/3.
When dealing with whole numbers, find the reciprocal by placing the number under 1.
For example:

• The reciprocal of 5 is 1/5.
• The reciprocal of -7 is -1/7.

The reciprocal is essential when dividing fractions.
For instance, to divide by a fraction, you multiply by its reciprocal.
The reciprocal ensures that multiplication and division within mathematics become more straightforward and manageable.