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Chapter 0: Problem 37

Give (a) the additive inverse and (b) the absolute value of each number. $$ 6 $$

Short Answer

Expert verified
The additive inverse is -6, and the absolute value is 6.

Step by step solution

01

Identify the number

The number given is 6.
02

Find the additive inverse

The additive inverse of a number is what you add to the number to get zero. For the number 6, its additive inverse is -6 because 6 + (-6) = 0.
03

Find the absolute value

The absolute value of a number is its distance from zero on the number line, without considering direction. For the number 6, the absolute value is 6 since it is already positive.

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Key Concepts

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

Additive Inverse
To understand the idea of an additive inverse, we must first know that this concept is about getting to zero. The additive inverse of a number is what you need to add to that number so the result is zero. It's like hitting a reset button back to zero.

For example, if we take the number 6, its additive inverse is -6. When you add 6 and -6 together, you get zero: \(6 + (-6) = 0\).

In general terms, if you have a number 'a', the additive inverse is '-a'.
  • For 8, the additive inverse is -8.
  • For -3, the additive inverse is 3.
  • For 0, the additive inverse is still 0!
It's always about pairing a number with its opposite to hit zero!
Absolute Value
The absolute value of a number is a measure of how far that number is from zero on the number line, ignoring which direction you go. Essentially, it disregards whether the number is positive or negative.

For instance, the absolute value of 6 is 6 because it is 6 units away from zero. Similarly, the absolute value of -6 is also 6 because -6 is 6 units away from zero.

The absolute value is often represented using vertical bars: \( |a| \). So, \( |6| = 6 \) and \( |-6| = 6 \).
  • For the number 7, the absolute value is 7.
  • For the number -5, the absolute value is 5.
  • For 0, the absolute value is 0.
Remember, absolute value is always non-negative because you're measuring distance, and distance can't be negative!
Number Line
The number line is a simple way to visualize numbers and their relationships. It's like a long ruler stretching infinitely in both directions.

In the middle of the number line, you have 0. Positive numbers are to the right of zero, and negative numbers are to the left. This makes it easy to see the distance of any number from zero, helping you easily find absolute values.
  • 6 is to the right of 0.
  • -3 is to the left of 0.
The number line also helps you visualize the concept of additive inverse. To find the additive inverse of 4, you look 4 units to the left of zero, landing at -4. Similarly, the additive inverse of -2 is 2 because you go 2 units to the right. This visual tool makes understanding these concepts much simpler!

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