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The Laws of Algebra

Suppose that x, y, and z are numbers.

The Commutative Laws

The Commutative Law of Addition

images/laws21.png

This laws says that the order in which you add two numbers does not matter.

The Commutative Law of Multiplication

images/laws22.png

This laws says that the order in which you multiply two numbers does not matter.

The Associative Laws

Only two numbers can be added or multiplied in one step.  The associative laws give us rules to add or multiply more than two numbers.

The Associative Law of Addition

images/laws23.png

If we want to add three numbers, we could add the second and third number and add that results to the first number.  Or we could add the first and second number and add the third number to that result.  This law tells us that we get the same number either way.  Effectively, this means that we do not have to write the parentheses to force the order that we do the arithmetic.

The Associative Law of Multiplication

images/laws24.png

If we want to multiply three numbers, we could multiply the second and third number and multiply that results to the first number.  Or we could multiply the first and second number and multiply the third number to that result.  This law tells us that we get the same number either way.  Effectively, this means that we do not have to write the parentheses to force the order that we do the arithmetic.

The Distributive Law

images/laws25.png

This law provides the relationship between addition and multiplication.

The Identity Properties

The Zero Property

There exists a number, images/laws26.png , such that images/laws27.png

The number zero is sometimes called the additive identity. Adding zero to a number results in that number.

The Multiplicative Identity Property

There exists a number, images/laws28.png , such that for any number images/laws29.png , images/laws210.png Multiplying one by a number results in that number.

To be complete, we state also that images/laws211.png .

Inverse Properties

The Additive Inverse

For any number images/laws212.png , there exists a number x such that images/laws213.png

For any number, there is minus that number.

The Multiplicative Inverse

If images/laws214.png is any number except 0, there exists a number x such that images/laws215.png

For any number except 0, there is a fraction, one divided by that number.

Laws of Equality

The Reflexive Property:

images/laws216.png

A number is equal to itself.

The Symmetric Property:

If images/laws217.png , then images/laws218.png
Equality works to both directions.

The Transitive Property:

If images/laws219.png , and images/laws220.png , then images/laws221.png

If images/laws222.png , then images/laws223.png
Adding equals to equals results in equals.

If images/laws224.png , then images/laws225.png
Multiplying equals by equals results in equals.

Laws of Inequality

If images/laws226.png , then images/laws227.png
Adding any constant to an inequality preserves the inequality.

If images/laws228.png and images/laws229.png , then images/laws230.png
Multiplying an inequality by a positive number preserves the inequality.

If images/laws231.png and images/laws232.png , then images/laws233.png
Multiplying an inequality by a negative number reverses the inequality.

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