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Introduction to Arithmetic Operations
Algebra Textbooks Boundless Algebra Numbers and Operations Introduction to Arithmetic Operations
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Concept Version 3
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Negative Numbers

Arithmetic operations can be performed on negative numbers according to specific rules.

Learning Objective

  • Calculate the sum, difference, product, and quotient of negative whole numbers


Key Points

    • The addition of two negative numbers results in a negative; the addition of a positive and negative number produces a number that has the same sign as the number of larger magnitude. 
    • Subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude, while subtracting a negative number yields the same result as adding a positive number.
    • The product of one positive number and one negative number is negative, and the product of two negative numbers is positive.
    • The quotient of one positive number and one negative number is negative, and the quotient of two negative numbers is positive.

Full Text

The Four Operations

Addition

The addition of two negative numbers is very similar to the addition of two positive numbers. For example:

$(−3) + (−5)  =  −8$

The underlying principle is that two debts—negative numbers— can be combined into a single debt of greater magnitude.

When adding together a mixture of positive and negative numbers, another way to write the negative numbers is as positive quantities being subtracted. For example:

$8 + (−3)  =  8 − 3  =  5$

Here, a credit of 8 is combined with a debt of 3, which yields a total credit of 5. However, if the negative number has greater magnitude, then the result is negative:

$(−8) + 3  =  3 − 8  =  −5 $

Similarly:

$(−2) + 7  =  7 − 2  =  5$

Here, a debt of 2 is combined with a credit of 7. The credit has greater magnitude than the debt, so the result is positive. But if the credit is less than the debt, the result is negative:

$2 + (−7)  =  2 − 7  =  −5$

Subtraction

Subtracting positive numbers from each other can yield a negative answer. For example, subtracting 8 from 5:

$5 − 8  =  −3 $

Subtracting a positive number is generally the same as adding the negative of that number. That is to say:

$5 − 8  =  5 + (−8)  =  −3 $

and

$(−3) − 5  =  (−3) + (−5)  =  −8$

Similarly, subtracting a negative number yields the same result as adding the positive of that number. The idea here  is that losing a debt is the same thing as gaining a credit. Therefore:

$3 − (−5)  =  3 + 5  =  8 $

and

$(−5) − (−8)  =  (−5) + 8  =  3$

Multiplication

When multiplying positive and negative numbers, the sign of the product is determined by the following rules:

  • The product of two positive numbers is positive.The product of one positive number and one negative number is negative.
  • The product of two negative numbers is positive.

For example:

$(−2) × 3  =  −6 $

This is simply because adding −2 together three times yields −6:

$(−2) × 3  =  (−2) + (−2) + (−2)  =  −6$

However,

$(−2) × (−3)  =  6$

The idea again here is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:

$\left (−2\text{ debts } \right ) \times \left ( −3 \text{ each} \right )  =  +6\text{ credit}$

Division

The sign rules for division are the same as for multiplication. 

  • Dividing two positive numbers yields a positive number.
  • Dividing one positive number and one negative number yields a negative number.
  • Dividing two negative numbers yields a positive number.

If the dividend and the divisor have the same sign, that is to say, the result is always positive. For example:

$8 ÷ (−2)  =  −4$

and

$(−8) ÷ 2  =  −4$

but

$(−8) ÷ (−2)  =  4$.

Additional Considerations

The basic properties of addition (commutative, associative, and distributive) also apply to negative numbers. For example, the following equation demonstrates the distributive property: 

$-3(2 + 5) = (-3)\cdot 2 + (-3)\cdot 5 $

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