In this section, we will learn how to operate with signed numbers.

 

A signed number has a weight and a sign.

For example: -5 has weight 5 and a negative sign “-“; 7 has weight 7 and a positive sign “+” which is usually omitted. 

Two numbers are opposite if they have the same weight but opposite signs. 

For example: the opposite of -5 is 5, and the opposite of 7 is -7. 

 

Addition: 

  • To add two numbers with the same sign, add their weights and place the result after the sign.
  • To add two numbers of opposite signs, find the difference of their weights (largest-smallest) and place the result after the sign of the number with the largest weight.

Examples: 

a)  $ -8 + 19 = 11$

b)   $-8 + 4 = -4$

c)   $6 + (-9) = -3$

d)  $7 + (-2) = 5$

e)  $-4 + (-7) = -11$

 

Subtraction (as Addition of the Opposite): 

  • To subtract a signed number is to add the opposite of that number, that is, change the subtraction sign to addition sign, change the number to its opposite, then add.  

Examples: 

a)  $3 – 7  = 3 + (-7) = -4$

b)   $-4 – 1 = -4 + (-1) = -5$

c)   $-4 – (-9) = -4 + 9 = 5$

d)  $-3 – 7 + 5 + 7 + 13 – 6 – (-9) = -3 + (-7) + 5 + 7 + 13 + (-6) + 9 = 18$

 

Multiplication: 

  • If the signs of the two numbers are the same, then the sign of the product is positive.
  • If the signs of the two numbers are different, then the sign of the product is negative.

Examples: 

a)   $(-5)(-8) = 40$

b)   $-6 \cdot 7  = -42$

c)   $(-3)(-5)\cdot 4(-2) = 15\cdot 4(-2) = 60(-2) = -120$

A positive exponent represents the number of times a number (known as base) is multiplied by itself.

Examples: 

a)   $5^2 = 5\cdot 5 = 25$

b)   $(-4)^3  = (-4)(-4)(-4) = -64$

c)   $-2^4 = – (2)(2)(2)(2) = -16$   Note: The base here is  2 not −2!

The exponential rules will be further discussed in Section 3.1. 

 

Division: 

  • If the signs of the two numbers are the same, then the sign of the quotient is positive.
  • If the signs of the two numbers are different, then the sign of the quotient is negative.

Examples: 

a)   $(-42)\div 7 = \displaystyle\frac{-42}{7} = -6$

b)   $81\div (-9) = \displaystyle\frac{81}{-9} = -9$

c)   $(-35)\div (-7) = \displaystyle\frac{-35}{-7} = 5$

d)   $0 \div 5 = \displaystyle\frac{0}{5} = 0$   Note:  When dividing 0 by any nonzero number,  the answer is always 0.

e)   $-10 \div 0 = \displaystyle\frac{-10}{0} =$undefined.   Note: Any number divided by 0 is undefined!

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Practice Problems: 

(a)   $-9 – 2$

(b)   $10 + (-3)$

(c)   $-10 – (-3)$

(d)   $(-6)^1$

(e)   $-15\cdot 6$

(f)  $\displaystyle\frac{-21}{-7}$

 

Answer Key: (a)    $-11$     (b)   $7$      (c)     $-7$       (d)     $-6$        (e)     $-90$        (f)     $3$

 

For more detailed explanation, please read: Arithmetic|Algebra Chapter 1.