Sovel Pre-Algebra Class

- x - = +

In mathematics, we can use the term 'rule', but in reality, there are no rules, only demonstrtions and proofs. Therefore, we must be able to demonstrate and prove each mathematical statement and expression before we can apply the tag 'rule' to it.

Listed below are five such demonstrations that discuss and illustrate the ideas of multiplying and dividing positive and negative values..

 An Inductive Method Number Line/Opposite of Division Property of One Reciprocals Positive and Negative Exponents return to homework page
Remember, a proof can only build upon a foundation of other related proofs

An Inductive approach1 to demonstrating
that a negative multiplied by a negative equals a positive

+ · + = +

We normally work in a world of all positives. Notice the pattern of change as we decrease the multiplier

 +10 +10 +10 +10 +10 +10 x 5 x +4 x +3 x +2 x +1 x 0 + 50 + 40 + 30 + 20 + 10 0

+ · - = -

If we continue this pattern, notice what happens to the product.

 +10 +10 +10 +10 +10 x -1 x -2 x -3 x -4 x -5 - 10 - 20 - 30 - 40 - 50

- · + = -

We have demonstrated that a + · - = -; notice that a -· + = - is also true.

 -10 -10 -10 -10 -10 -10 x 5 x +4 x +3 x +2 x +1 x 0 -50 -40 -30 -20 -10 0

- · - = +

If we continue this pattern, notice what happens to the product.

 - 10 - 10 - 10 - 10 - 10 x -1 x -2 x -3 x -4 x -5 +10 +20 +30 +40 +50

1 Farrell, Margaret A. and Farmer, Walter A. Secondary Mathematics Instruction: An Integrated Approach. Janson Publications, Inc.: Providence, R.I., 1995: pages 94-96.

The Number Line and Opposites

Understanding Integer opposites can also demonstrate a negative times a negative equals a positive

 an integer it's opposite it's opposite 5 -5 -(-5) + - +

The illustration above can be demonstrated visually by the tracking of these integers on a number line.

Division Property of One

Other Properties of real numbers can also demonstrate that a negative divided by a negative can equal a positive, such as the Division Property of One.

This can be stated as,

for every non-zero number a,

a ÷ 1 = a and a ÷ a = 1, such that any value divided by itself will equal a positive 1;

therefore, if a is positive, a positive divided by a positive equals a positive 1; and

if a is a negative value, a negative divided by a negative must, therefore, also equal a positive 1

Reciprocal

Certain mathematical definitions can also be used to demonstrate that a negative times a negative equals a positive. For example,

two numbers, like , whose product is 1 are called reciprocals.

The reciprocal of because = 1. Every non-zero rational number has exactly one reciprocal.

Therefore, the following negative number, and its reciprocal, also produce the product +1:

Positive and Negative Exponents

 coefficient --> 4 x 2 <-- exponent variable

The exponent rests on a base. When two terms have the same base and are multiplying each other, you may add the exponents [as a shortcut to simplifying]. For example:

 (4 2) (4 4) = 4 2 + 4 = 4 6 OR (4 5) (4 -3) = 4 5 + -3 = 4 2

It is important to remember that while the base is a factor, the exponent is not. The exponent simply tells us the number of times the base multiplies itself, as a factor.

The path path for converting a negative exponent to a positive exponent

 x 4 = (1) (x) (x) (x) (x) = x 4 4 4 = (1) (4) (4) (4) (4) = 256 x 3 = (1) (x) (x) (x) = x 3 4 3 = (1) (4) (4) (4) = 64 x 2 = (1) (x) (x) = x 2 4 2 = (1) (4) (4) = 16 x 1 = (1) (x) = x 1 4 1 = (1) (4) = 4 x 0 = 1 = 1 4 0 = 1 = 1 x -1 = (1) (1/x 1) = 1/x 1 4 -1 = (1) (1/4 1) = 1/4 x -2 = (1) (1/x 2) = 1/x 2 4 -2 = (1) (1/4 2) = 1/16 x -3 = (1) (1/x 3) = 1/x 3 4 -3 = (1) (1/4 3 ) = 1/64

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