Problem 55 - Prealgebra

Question

Give the opposite of each of the following numbers. $$75$$

Step-by-Step Solution

Answer

The opposite of 75 is -75.

Step 1: Understanding Opposite Numbers

Opposite numbers, in mathematics, are numbers that have the same magnitude but opposite signs. Therefore, the opposite of a positive number is its negative equivalent.

Step 2: Apply the Concept of Opposite to 75

Given the number 75, which is positive, its opposite would be a number of the same magnitude but negative sign. Hence, the opposite of 75 is -75.

Key Concepts

Positive and Negative Numbers

Positive and negative numbers are two fundamental types of numbers in mathematics, closely related to the number line. A positive number is a number greater than zero and is usually written without a plus sign, though it can be included; for example, both "5" and "+5" represent the same positive integer. On a number line, you move to the right to denote positive numbers. These numbers signify values above the reference point of zero, like gains, profits, or elevations.

Negative numbers, on the other hand, are less than zero and are typically preceded by a minus sign, such as "-5". They appear to the left of zero on the number line and are used to describe values that are below the reference point, such as debts, losses, or depths beneath sea level.

Both positive and negative numbers help us understand real-world situations where direction and balance matter, such as temperature changes or accounting balances.

Integer Operations

Integer operations involve mathematical actions using whole numbers, which include both positive and negative numbers, as well as zero. There are several key operations that you should become familiar with when working with integers:

• Addition: When adding two positive integers, you simply find their sum as you usually would. When adding a positive and a negative integer, you subtract the smaller magnitude from the larger and keep the sign of the larger absolute value.
• Subtraction: Subtracting integers can be understood as adding the opposite. To subtract an integer, you add its opposite. Thus, \(5 - (-3)\) becomes \(5 + 3\), which equals 8.
• Multiplication: When multiplying two integers with the same sign (both positive or both negative), the product is positive. If the signs are different, the product is negative; for example, \(-2 \times 3 = -6\) and \(-2 \times -3 = 6\).
• Division: Like multiplication, if two integers have the same sign, the quotient is positive. If they have different signs, the quotient is negative.

Understanding these operations is essential for solving more complex math problems involving integers.

Mathematical Signs

Mathematical signs are symbols used to represent operations or indicate the nature of a number. The most common signs include:

• Plus Sign (+): Indicates addition or a positive number. For example, \(7 + 3\) implies adding 3 to 7, and "+5" signals a positive five.
• Minus Sign (-): Used for subtraction or to denote negative numbers. So, \(10 - 4\) means 4 is subtracted from 10, and "-8" denotes negative eight.
• Multiplication Sign (× or *): Shows the operation of multiplying one number by another, like \(4 \times 2 = 8\). The sign can vary, but the operation remains consistent across different notations.
• Division Sign (÷ or /): Represents dividing one number by another, such as \(9 \div 3 = 3\).

Learning to recognize and properly use these signs is critical in mathematical operations as they convey specific actions to take with numbers, guiding the process in problem-solving tasks.