✓ (–a) – (–b) = (–a) + (+b)
The following examples put the process of subtracting signed numbers into real-life terms:
✓ The submarine was 60 feet below the surface when the skipper shouted, “Dive!” It went down another 40 feet:
– 60 – (+40) = –60 + (–40) = –100
Change from subtraction to addition. Change the 40 to its opposite, –40. Then use the addition rule. The submarine is now 100 feet below the surface.
✓ Some kids are pretending that they’re on a reality-TV program and clinging to some footholds on a climbing wall. A team challenges the position of the opposing team’s player. “You were supposed to go down 3 feet, then up 8 feet, then down 4 feet. You shouldn’t be 1 foot higher than you started!” The referee decides to check by having the player go backward – do the opposite moves. Making the player do the opposite, or subtracting the moves:
– (–3) – (+8) – (–4) = +(+3) + (–8) + (+4) = –5 + (+4) = –1
The player ended up 1 foot lower than where he started, so he had moved correctly in the first place.
Here are some examples of subtracting signed numbers:
✓ –16 – 4 = –16 + (–4) = –20: The subtraction becomes addition, and the +4 becomes negative. Then, because you’re adding two signed numbers with the same sign, you find the sum and attach their common negative sign.
✓ –3 – (–5) = –3 + (+5) = 2: The subtraction becomes addition, and the –5 becomes positive. When adding numbers with opposite signs, you find their difference. The 2 is positive because the +5 is farther from 0.
✓ 9 – (–7) = 9 + (+7) = 16: The subtraction becomes addition, and the –7 becomes positive. When adding numbers with the same sign, you find their sum. The two numbers are now both positive, so the answer is positive.
Remember: To subtract two signed numbers:
Examples
Q. (–8) – (–5) =
A. Change the problem to (–8) + (+5) =. The answer is –3.
Q. 6 – (+11) =
A. Change the problem to 6 + (–11) =. The answer is –5.
Practice Questions
1. 5 – (–2) =
2. –6 – (–8) =
3. 4 – 87 =
4. 0 – (–15) =
5. 2.4 – (–6.8) =
6. –15 – (–11) =
Practice Answers
1. 7.
2. 2.
3. – 83.
4. 15.
5. 9.2.
6. – 4.
When you multiply two or more numbers, you just multiply them without worrying about the sign of the answer until the end. Then to assign the sign, just count the number of negative signs in the problem. If the number of negative signs is an even number, the answer is positive. If the number of negative signs is odd, the answer is negative.
Remember: The product of two signed numbers:
The product of more than two signed numbers:
(+)(+)(+)(–)(–)(–)(–) has a positive answer because there are an even number of negative factors.
(+)(+)(+)(–)(–)(–) has a negative answer because there are an odd number of negative factors.
Examples
Q. (–2)(–3) =
A. There are two negative signs in the problem. The answer is +6.
Q. (–2)(+3)(–1)(+1)(–4) =
A. There are three negative signs in the problem. The answer is –24.
Practice Questions
1. (–6)(3) =
2. (14)(–1) =
3. (–6)(–3) =
4. (6)(–3)(4)(–2) =
5. (–1)(–1)(–1)(–1)(–1)(2) =
6. (–10)(2)(3)(1)(–1) =
Practice Answers
1. – 18. The multiplication problem has one negative, and 1 is an odd number.
2. – 14. The multiplication problem has one negative, and 1 is an odd number.
3. 18. The multiplication problem has two negatives, and 2 is an even number.
4. 144. The multiplication problem has two negatives.
5. – 2. The multiplication problem has five negatives.
6. 60. The multiplication problem has two negatives.
The rules for dividing signed numbers are exactly the same as those for multiplying signed numbers – as far as the sign goes (see “Multiplying Signed Numbers” earlier in this chapter.) The rules do differ though
