Arithmetic Of Negative Numbers

The concept of negative numbers is not as straight forward as the concept of positive numbers. One can easily conceptualize positive quantities. But the conceptualization of negative quantities is somewhat difficult. One excellent way to approach the arithmetic of negative numbers is to see them as debts, deficits , liabilities or measurements below a reference point (e.g ground). The arithmetic of negative numbers becomes easier with such visualizations.

Addition of two negative numbers
Example: -3 + -3. Interpretation: 3 units of debt is added to 3 units of debt. Result is 6 units of debt. So, -3 + -3 = -6. Often the addition operator (+) is omitted. Thus -3 - 3 = -6.

A positive number added to a negative number
Example: -7 + 4. Interpretation: 7 units of debt is reduced by a payment of 4 units. Result is 3 units of debt. So, -7 + 4 = -3. The reverse which is 4 + -7 = 4 - 7 = -3, is the case where a debt of 7 units is added to a positive account of 4 units.

A negative number subtracted from a negative number
Example: -9 - (-5). Interpretation: a debt of 5 units is removed from a debt of 9 units. Result is 4 units of debt. So, -9 -(-5) = -4. The removal of debt is due to either a payment on debt, or an adjustment such as the cancellation of debt. So, -9 -(-5) = -9 + 5 = -4. In general, when the subtraction operator operates on a negative number, it becomes a positive number. The expression is then handled as an addition of a negative number and a positive number.

A positive number subtracted from a negative number
Example: -8 - (+7). Interpretation: an overdraft. In other words, a debtor already oweing 8 units, borrowed an additional 7 units. Result is 15 units of debt. So, -8 - (+7) = -15. Often the positive sign of the positive number is omitted. Thus -8 - (+7) = -8 - 7 = -15.

A negative number multiplied by a positive number
Example: ( -7) x (+5). Interpretaion: A debt of 7 units has been incurred 5 times, that is, -7 + -7 + -7 + -7 + -7. Result is 35 units of debt. So,
(-7) x (+5) = -35. In general, the multiplication of a negative number by a positve number results in a negative number with a numeric value equal to the product of the numbers involved in the multiplication.

A negative number multiplied by a negative number
Example: (-7) x (-5). Interpretation: A debt of 7 units has been removed 5 times. As previously stated, a removal of debt of x units which is -(-x), translates to (+x). So, (-7) x (-5) = -(-7) -(-7) -(-7) -(-7) -(-7) = +7 +7 +7 +7 +7 = 35. In general, the multiplication of a negative number by a negative number results in a positive number with a numeric value equal to the product of the multiplicands.

A negative number divided by a negative number
Divisions involving negative numbers are straight forward, once the multiplications involving negative numbers are understood. This is because the numerator (dividend) of a division equals the product of the denominator (divisor) and the quotient . Example: (-6)/(-2). Interpretation: how many times is a debt of 2 units contained in a debt of 6 units? The answer is 3 times. We expect the sign of the quotient to be positive because the product of the quotient and the divisor must equal the dividend. So, (-6)/(-2) = 3. In general, when a negative number is divided by a negative number, the quotient is a positive number.

A negative number divided by a positive number
Example: (-9)/(+3). Interpretation: what is the debt per time-unit if a debt of 9 units has been incurred after 3 time-units (a time-unit may be an hour, or a day , or a year) ? The answer is 3 units of debt per time-unit. We expect the sign of the quotient to be negative because the product of the quotient and the divisor must equal the dividend. So, (-9)/(+3) = -3. In general, when a negative number is divided by a positive number, the quotient is a negative number.

A positive number divided by a negative number
Example: (+21)/(-7). Interpretation: how many times can a debt of 7 units be removed to obtain a positive value of 21 units. The answer is three. Since this is a removal of debt and since the product of the quotient and divisor must equal the dividend, we expect the sign to the answer to be negative. So, (+21)/(-7) = -3. In general, when a positive number is divided by a negative number, the quotient is a negative number.

Negative numbers play important roles in mathematics. The interpretation of negative numbers is usually contextual. Nonetheless, their role in measurement almost always represent the concept of deficit, that is, the concept of being less than zero.

Peter Oye Sagay

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