Problem 19 - Prealgebra

Question

Find the multiplicative inverse of each number. $$-7$$

Step-by-Step Solution

Answer

The multiplicative inverse of -7 is -1/7.

Step 1: Understand the Multiplicative Inverse

The multiplicative inverse of a number is a number that, when multiplied by the original number, results in 1. If the number is represented as 'a', the multiplicative inverse is '1/a'.

Step 2: Identify the Number

The number given is -7. We want to find a number which, when multiplied by -7, gives us 1.

Step 3: Apply the Reciprocal

The reciprocal of a number '-a' is '-(1/a)'. Therefore, the reciprocal of -7 is -1/7, because when -7 is multiplied by -1/7, the product is 1.

Step 4: Verification

Multiply -7 by its reciprocal, -1/7, to ensure it equals 1. Calculate: \(-7 \times -\frac{1}{7} = \frac{7}{7} = 1\). The operation confirms we have found the correct multiplicative inverse.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (-) in front of the number. Negative numbers are often used to represent values like debts, temperatures below zero, or elevations below sea level.
When working with negative numbers, it's important to note that two negatives make a positive. For instance, multiplying or dividing two negative numbers results in a positive value.

• When you add a negative number, it's similar to subtraction. Example: 3 + (-2) equals 1.
• When subtracting a negative number, it’s like adding the positive equivalent. Example: 3 - (-2) equals 5.

Elaborating on negative numbers can help in understanding their behavior in various mathematical operations, such as finding multiplicative inverses.

What is a Reciprocal?

The reciprocal of a number is very important in algebra and arithmetic. Simply put, the reciprocal of a number is 1 divided by that number. If you have a number 'a', its reciprocal is expressed as \( \frac{1}{a} \).
Reciprocals are particularly useful when you wish to find the multiplicative inverse. For example:

• The reciprocal of 5 is \( \frac{1}{5} \).
• The reciprocal of -3 is \( -\frac{1}{3} \).

Understanding reciprocals is also critical when working with fractions and division problems, as they allow you to switch division into multiplication.

Reciprocals and Fractions

Reciprocals are directly linked to fractions. The reciprocal of a fraction \( \frac{a}{b} \) is simply \( \frac{b}{a} \). In this sense, every whole number can be seen as a fraction with a denominator of 1, making it easier to find their reciprocals.

Multiplication Basics and Beyond

Multiplication is a core mathematical operation representing repeated addition. In simpler terms, multiplying two numbers gives the total of one number added to itself a specified number of times.
For example, multiplying 3 by 4 tells you to add 3 to itself four times: 3 + 3 + 3 + 3, which equals 12.
Multiplication is used to calculate areas, volumes, and in various algebraic manipulations.

• Key property of multiplication: the commutative property (a * b = b * a)
• The identity element is 1, as any number multiplied by 1 remains unchanged.
• Zero property: any number multiplied by 0 results in 0.

Negative Numbers in Multiplication

When multiplying negative numbers, remember:

• A negative times a positive results in a negative.
• A negative times a negative results in a positive.
• Example: \(-5 \times -4 = 20\)

These principles form the basis for many areas of math, including prealgebra.

Introduction to Prealgebra

Prealgebra is a foundational course that prepares students for Algebra. It covers basic arithmetic and introduces crucial concepts such as variables, factors, basic equations, and proportions.
For students just starting with mathematics, prealgebra serves as an essential framework. It builds a strong base by combining elements of arithmetic with simple algebraic concepts.

Key Concepts in Prealgebra

Prealgebra focuses on:

• Understanding whole numbers, fractions, decimals, and negative numbers.
• Grasping basic operations—addition, subtraction, multiplication, and division—of numbers.
• Introducing basic concepts of variables and how they can be used to form equations.
• Learning to solve simple equations and inequalities.

With a mastery of prealgebra, students can smoothly transition into more advanced math classes with confidence.