Mark Zegarelli - Basic Math & Pre-Algebra All-in-One For Dummies (+ Chapter Quizzes Online)

In contrast to negation, placing two bars around a number gives you the absolute value of that number. Absolute value is the number’s distance from 0 on the number line — that is, it’s the positive value of a number, regardless of whether you started out with a negative or positive number:

The absolute value of a positive number is the same number.

The absolute value of a negative number makes it a positive number.

Placing absolute value bars around 0 doesn’t change its value, so .

Placing a negative sign outside absolute value bars gives you a negative result — for example, .

Q.Negate the number 7.

A. −7.Negate 7 by attaching a negative sign to it: −7.

Q.Find the negation of −3.

A. 3.The negation of . The two adjacent negative signs cancel out, which gives you 3.

Q.What’s the negation of ?

A. 5.First, do the subtraction, which tells you . To find the negation of −5, attach a negative sign to the answer: . The two adjacent negative signs cancel out, which gives you 5.

Q.What number does equal?

A. 9.The number 9 is already positive, so the absolute value of 9 is also 9.

Q.What number does equal?

A. 17.Because −17 is negative, the absolute value of .

Q.Solve this absolute value problem:

A. −4.Do the subtraction first: , which is negative, so the absolute value of . But the negative sign on the left (outside the absolute value bars in the original expression) negates this result, so the answer is −4.

3 Negate each of the following numbers and expressions by attaching a negative sign and then canceling out negative signs when possible:

(a)

(b)

(c)

(d)

(e)

(f)

4 Solve the following absolute value problems:

(a)

(b)

(c)

(d)

(e)

(f)

The great secret to adding and subtracting negative numbers is to turn every problem into a series of ups and downs on the number line. When you know how to do this, you find that all these problems are quite simple.

So in this section, I explain how to add and subtract negative numbers on the number line. Don’t worry about memorizing every little bit of this procedure. Instead, just follow along so you get a sense of how negative numbers fit onto the number line. (If you need a quick refresher on how the number line works, see Chapter 1.)

When you’re adding and subtracting on the number line, starting with a negative number isn’t much different from starting with a positive number. For example, suppose you want to calculate . Using the up and down rule, you start at and go up :

Similarly, suppose you want to calculate . Again, the up and down rule helps you out. You’re subtracting, so move to the left: start at , down :