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Chapter 2: Problem 86

Give the opposite of each number. $$3$$

Short Answer

Expert verified
The opposite of 3 is -3.

Step by step solution

01

Understand Opposite

To find the opposite of a number, we need to identify what number, when added to the original, results in zero. This is often referred to as the 'additive inverse.' For any real number, the opposite is simply the negative of that number.
02

Find Opposite of Given Number

Given the number is 3, the opposite of 3 is found by multiplying 3 by -1. Thus, the opposite of 3 is calculated as follows: \[-1 \times 3 = -3\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Opposite Numbers
When we talk about opposite numbers, we refer to numbers that are equal in distance from zero but lie on opposite sides on the number line. The opposite of any given number can be simply identified by changing its sign.

For example:
  • The opposite of positive 3 is negative 3, and the opposite of negative 3 is positive 3.
  • If you start at 3 on the number line and move to -3, you've moved 6 units, 3 to the left to get away from zero and 3 to reach the opposite.
Understanding opposite numbers is fundamental in many areas of math. They help in understanding balance as well as in solving equations and inequalities.
Opposite numbers always add up to zero鈥攁 key concept that links to the idea of additive inverse.
Additive Inverse
The term 'additive inverse' might sound complex, but it simply refers to a number that when added to your original number yields zero. In other words, the additive inverse of a number is its opposite.

Here's how you can visualize it:
  • Take any real number, like 5. Its additive inverse is -5 because 5 + (-5) = 0.
  • Even for negative numbers, this holds true. The additive inverse of -7 is 7, because -7 + 7 = 0.
In prealgebra and beyond, knowing each number's additive inverse helps when we need to "cancel out numbers" to simplify equations or solve for unknowns. It's a powerful concept because it reassures us that every number has a partner that returns us to zero.
Real Numbers
Real numbers are the foundation of prealgebra and encompass a broad set of numbers that include integers, fractions, and decimals. They represent every possible point on the number line.

Some key points about real numbers include:
  • They are split into rational numbers (like \( rac{1}{2}, 3, -5 \)) and irrational numbers (such as \( ext{蟺} \) or \( ext{鈭�2} \)).
  • Integers, both positive and negative, are a subset of real numbers.
  • Real numbers can be both whole numbers and numbers with fractional parts.
Understanding real numbers is crucial because they are used in everyday life for counting, measuring, and ordering. Every number we typically use fits within the real number system, which makes them essential for developing algebraic skills and navigating the mathematical landscape.

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Most popular questions from this chapter

Combine the following by using the rule for addition of positive and negative numbers. $$7+8$$

Draw a number line from 10 to 10 and use it to add the following numbers. $$10+(-6)$$

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$7-3-2$$

Perform the indicated operations. $$32 \div 4$$

In the metric system, the scale we use to measure temperature is the Celsius scale. On this scale water boils at 100 degrees and freezes at 0 degrees. When we write 100 degrees measured on the Celsius scale, we use the notation \(100^{\circ} \mathrm{C},\) which is read "100 degrees Celsius." If we know the temperature in degrees Fahrenheit, we can convert to degrees Celsius by using the formula $$c=\frac{5(F-32)}{9}$$ where \(F\) is the temperature in degrees Fahrenheit. Use this formula to find the temperature in degrees Celsius for each of the following Fahrenheit temperatures. \(41^{\circ} \mathrm{F}\)
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