Problem 58 - Prealgebra

Question

Give the opposite of each of the following numbers. $$1$$

Step-by-Step Solution

Answer

The opposite of 1 is -1.

Step 1: Understanding Opposite Numbers

The opposite of a number is the number that, when added to the original number, sums to zero. This is also known as the additive inverse.

Step 2: Determine the Opposite of 1

The opposite of a positive number is a negative number with the same absolute value. Therefore, the opposite of \(1\) is \(-1\).

Step 3: Verify by Addition

To confirm the opposite, add \(1\) and its opposite \(-1\). The result should be zero: \(1 + (-1) = 0\). This verifies that \(-1\) is indeed the opposite of \(1\).

Key Concepts

Additive Inverse

The concept of additive inverse might sound complex at first, but it's actually quite simple. Imagine you have a number, say 1. The additive inverse of this number is the number that you can add to 1 to get zero as a result. It is like finding a partner for a number that balances it out to zero.

For any number, the additive inverse is just the opposite sign of the number. Thus, for the number 1, its additive inverse is -1.

If you ever wonder whether you found the right additive inverse, simply add the number and its assumed inverse. If they sum to zero, you've got it right! For example:

• For 1: 1 + (-1) = 0
• For -3: -3 + 3 = 0

The additive inverse is an essential concept in mathematics, helping us understand how numbers balance each other and relate within equations. Remember, it is often called the opposite number for a reason!

Absolute Value

The absolute value of a number is another fundamental mathematical concept that helps us in many calculations. Absolute value refers to how far a number is from zero on the number line, ignoring in which direction. This means that absolute value is always a positive number or zero.

Consider this:

• The absolute value of 1 is 1.
• The absolute value of -1 is still 1.
• The absolute value of 0 is simply 0.

What makes absolute value really useful is its ability to measure magnitude without concern for direction. Think of it like the distance between a number and zero, whether you're moving left or right on the number line doesn't matter.

So, when you need to find the opposite numbers or work with distances in math, absolute value is your friend, making sure you only focus on the size, not the sign!

Integer Operations

Integer operations are the basic math processes involving whole numbers. These include addition, subtraction, multiplication, and division of integers. Understanding these operations is crucial because integers are used throughout all areas of math.

When dealing with integers, remember the rules for combining positive and negative numbers:

• Adding two positive numbers gives a positive sum.
• Adding two negative numbers gives a negative sum.
• Adding a positive and a negative number requires comparing their absolute values. The sum takes the sign of the larger absolute value.
• Subtracting is like adding a negative. For example, 5 - 3 is the same as 5 + (-3).

Multiplication and division follow similar rules:

• Multiplying two numbers with the same sign gives a positive product.
• Multiplying two numbers with different signs gives a negative product.
• For division, the rules are the same as multiplication: like signs mean positive, unlike signs mean negative.

Mastering integer operations helps in understanding how numbers interact with each other, a crucial skill for solving real-world problems and further math studies.