Sovel PreAlgebra 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 approach^{1}
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 9496.
Understanding Integer opposites can also demonstrate a negative times a negative equals a positive









The illustration above can be demonstrated visually by the tracking of these integers on a number line.
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 nonzero 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 nonzero rational number has exactly one reciprocal.
Therefore, the following negative number, and its reciprocal, also produce the product +1:
coefficient > 

< exponent 

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:



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.
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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For questions, suggestions or corrections, you may contact Barry Sovel
Barry Sovel 1996