Problem 30 - Algebra 2

Question

Identify the additive inverse and multiplicative inverse for each number. $$ -0.125 $$

Step-by-Step Solution

Answer

The additive inverse of \(-0.125\) is \(0.125\) and the multiplicative inverse is \(-8\).

Step 1: Define Additive Inverse

The additive inverse of a number is a number that, when added to the original number, results in zero. For any real number \(x\), the additive inverse is \(-x\).

Step 2: Find the Additive Inverse of \(-0.125\)

To find the additive inverse of \(-0.125\), we need to find a number that when added to \(-0.125\) gives zero. Mathematically, this number is \(0.125\) because: \(-0.125 + 0.125 = 0\).

Step 3: Define Multiplicative Inverse

The multiplicative inverse (or reciprocal) of a number is a number that, when multiplied by the original number, results in one. For any non-zero real number \(x\), the multiplicative inverse is \(\frac{1}{x}\).

Step 4: Find the Multiplicative Inverse of \(-0.125\)

To find the multiplicative inverse of \(-0.125\), we calculate \(\frac{1}{-0.125}\). Therefore, \[\frac{1}{-0.125} = -8\]. This is because: \(-0.125 \times -8 = 1\).

Step 5: Summary of Inverses

The additive inverse of \(-0.125\) is \(0.125\) and the multiplicative inverse is \(-8\).

Key Concepts

Additive Inverse

The additive inverse is a simple yet crucial concept in mathematics. For any real number, the additive inverse is the value that brings the original number to zero when summed together.

• The additive inverse of a real number \(x\) is \(-x\).
• It always satisfies the equation: \(x + (-x) = 0\).

In our given problem, we have the number -0.125.To find its additive inverse, we look for a number that, when added to -0.125, results in zero.
The answer is straightforward: it's 0.125.This is because \(-0.125 + 0.125 = 0\). As you can see, the concept of an additive inverse helps us understand how numbers can form perfect balances with their opposites.

Multiplicative Inverse

A multiplicative inverse is also known as a reciprocal. This is the number that multiplies with a given number to yield one.

• For any non-zero real number \(x\), the multiplicative inverse is \(\frac{1}{x}\).
• This satisfies the equation: \(x \times \frac{1}{x} = 1\).

Multiplicative inverses are crucial when solving equations or simplifying expressions. For number -0.125, the goal is to find its reciprocal.
To do this, simply set up the division: \(-0.125\), which equals \(\frac{1}{-0.125}\).
On solving it through calculation, we find this equals to \(-8\).
This means when \(-0.125\) is multiplied by \(-8\), it results in 1, which confirms that -8 is indeed the multiplicative inverse.

Real Numbers

Real numbers are an essential part of mathematics, covering every number imaginable from the negative to positive infinity, including everything in between: integers, rational numbers, and irrational numbers.
These numbers can be represented on a number line, which helps visualize concepts like inverses. The concept of real numbers is significant when discussing additive and multiplicative inverses.

• For real numbers, the additive inverse always exists and is simply a number's negative counterpart.
• The multiplicative inverse is present for all non-zero real numbers and provides a foundation for many algebraic processes.

Real numbers allow mathematicians to explore and understand the properties of numbers within a continuous framework, making them an indispensable part of arithmetic and higher-level math.