1.2: Negative Numbers
- Page ID
- 56840
Negative numbers are a fact of life, from winter temperatures to our bank accounts. Let’s practice evaluating expressions involving negative numbers.
Absolute Value
The absolute value of a number is its distance from \(0\). You can think of it as the size of a number without identifying it as positive or negative. Numbers with the same absolute value but different signs, such as \(3\) and \(-3\), are called opposites. The absolute value of \(-3\) is \(3\), and the absolute value of \(3\) is also \(3\).
We use a pair of straight vertical bars to indicate absolute value; for example,\(|-3|=3\) and \(|3|=3\).
Evaluate each expression.
1. \(|-5|\)
2. \(|5|\)
- Answer
-
1. 5
2. 5
Adding Negative Numbers
To add two negative numbers, add their absolute values (i.e., ignore the negative signs) and make the final answer negative.
Perform each addition.
3. \(-8+(-7)\)
4. \(-13+(-9)\)
- Answer
-
3. -15
4. -22
To add a positive number and a negative number, we subtract the smaller absolute value from the larger. If the positive number has the larger absolute value, the final answer is positive. If the negative number has the larger absolute value, the final answer is negative.
Perform each addition.
5. \(7+(-3)\)
6. \(-7+3\)
7. \(14+(-23)\)
8. \(-14+23\)
9. The temperature at noon on a chilly Monday was \(-7\)°F. By the next day at noon, the temperature had risen \(25\)°F. What was the temperature at noon on Tuesday?
- Answer
-
5. 4
6. -4
7. -9
8. 9
9. \(18\)°F
If an expression consists of only additions, we can break the rules for order of operations and add the numbers in whatever order we choose.
Evaluate each expression using any shortcuts that you notice.
10. \(-10+4+(-4)+3+10\)
11. \(-291+73+(-9)+27\)
- Answer
-
10. \(3\)
11. -200
Subtracting Negative Numbers
The image below shows part of a paystub in which an $ \(18\) payment needed to be made, but the payroll folks wanted to track the payment in the deductions category. Of course, a positive number in the deductions will subtract money away from the paycheck. Here, though, a deduction of negative \(18\) dollars has the effect of adding\(18\) dollars to the paycheck. Subtracting a negative amount is equivalent to adding a positive amount.
To subtract two signed numbers, we add the first number to the opposite of the second number.
Perform each subtraction.
12. \(5-2\)
13. \(2-5\)
14. \(-2-5\)
15. \(-5-2\)
16. \(2-(-5)\)
17. \(5-(-2)\)
18. \(-2-(-5)\)
19. \(-5-(-2)\)
20. One day in February, the temperature in Portland, Oregon is \(43\)°F, and the temperature in Portland, Maine is \(-12\)°F. What is the difference in temperature?
- Answer
-
12. 3
13. -3
14. -7
15. -7
16. 7
17. 7
18. 3
19. -3
20. \(55\)°F difference
Multiplying Negative Numbers
Suppose you spend \(3\) dollars on a coffee every day. We could represent spending 3 dollars as a negative number, \(-3\) dollars. Over the course of a \(5\)-day work week, you would spend \(15\) dollars, which we could represent as \(-15\) dollars. This shows that \(-3\cdot5=-15\), or \(5\cdot-3=-15\).
If two numbers with opposite signs are multiplied, the product is negative.
Find each product.
21. \(-4\cdot3\)
22. \(5(-8)\)
- Answer
-
21. -12
22. -40
Going back to our coffee example, we saw that \(5(-3)=-15\). Therefore, the opposite of \(5(-3)\) must be positive \(15\). Because \(-5\) is the opposite of \(5\), this implies that \(-5(-3)=15\).
If two numbers with the same sign are multiplied, the product is positive.
WARNING! These rules are different from the rules for addition; be careful not to mix them up.
Find each product.
23. \(-2(-9)\)
24. \(-3(-7)\)
- Answer
-
23. 18
24. 21
Recall that an exponent represents a repeated multiplication. Let’s see what happens when we raise a negative number to an exponent.
Evaluate each expression.
25. \((-2)^2\)
26. \((-2)^3\)
27. \((-2)^4\)
28. \((-2)^5\)
- Answer
-
25. \(4\)
26. -8
27. 16
28. -32
If a negative number is raised to an odd power, the result is negative.
If a negative number is raised to an even power, the result is positive.
Dividing Negative Numbers
Let’s go back to the coffee example we saw earlier: \(-3\cdot5=-15\). We can rewrite this fact using division and see that \(-15\div5=-3\); a negative divided by a positive gives a negative result. Also, \(-15\div-3=5\); a negative divided by a negative gives a positive result. This means that the rules for division work exactly like the rules for multiplication.
If two numbers with opposite signs are divided, the quotient is negative.
If two numbers with the same sign are divided, the quotient is positive.
Find each quotient.
29. \(-42\div6\)
30. \(32\div(-8)\)
31. \(-27\div(-3)\)
32. \(0\div4\)
33. \(0\div(-4)\)
34. \(4\div0\)
- Answer
-
29. -7
30. -4
31. 9
32. 0
33. 0
34. undefined
Go ahead and check those last three exercises with a calculator. Any surprises?
- 0 divided by another number is 0.
- A number divided by 0 is undefined, or not a real number.
Here’s a quick explanation of why \(4\div0\) can’t be a real number. Suppose that there is a mystery number, which we’ll call \(n\), such that \(4\div0=n\). Then we can rewrite this division as a related multiplication, \(n\cdot0=4\). But because \(0\) times any number is \(0\), the left side of this equation is \(0\), and we get the result that \(0=4\), which doesn’t make sense. Therefore, there is no such number \(n\), and \(4\div0\) cannot be a real number.
Order of Operations with Negative Numbers
P: Work inside of parentheses or grouping symbols, following the order PEMDAS as necessary.
E: Evaluate exponents.
MD: Perform multiplications and divisions from left to right.
AS: Perform additions and subtractions from left to right.
Let’s finish up this module with some order of operations practice.
Evaluate each expression using the order of operations.
35. \((2-5)^2\cdot2+1\)
36. \(2-5^2\cdot(2+1)\)
37. \([7(-2)+16]\div2\)
38. \(7(-2)+16\div2\)
39. \(\dfrac{1-3^4}{2(5)}\)
40. \(\dfrac{(1-3)^4}{2}\cdot5\)
- Answer
-
35. \(19\)
36. -73
37. 1
38. -6
39. -8
40. 40