# CRAM Math Properties of Real Numbers

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For the first two semesters (Algebra I/II), nearly all work is done in the real number system. The following properties of real numbers provide us with a definition of real numbers in the context of algebra and CRAM Math.

There are other number systems in algebra, such as the natural numbers and the rational numbers, but for now, we will concentrate on learning the real numbers as the most general system usually taught in Algebra I/II.

## Two basic operations on real numbers

To start, we will allow that you can add and multiply real numbers together according to the following properties.

## Identity Property of Addition of Real Numbers

The sum of any real number a and zero is equal to a.

$a + 0 = a$

## Identity Property of Multiplication of Real Numbers

The product of one and any real number a is equal to a.

$1a = a$

One is the multiplicative identity.

As a practical matter, nobody ever writes "$1a$" instead of "$a$" as part of a final answer, and your teacher may mark an answer like that as not finished or just plain wrong.
Of course, if any variable has any coefficient other than precisely 1, then write it down!

## Closure Property of Addition of Real Numbers

The sum of two real numbers a and b is a real number.

$(a + b) \in \Re$

## Closure Property of Multiplication of Real Numbers

The product of two real numbers a and b is a real number.

$(ab) \in \Re$

## Commutative Property of Addition of Real Numbers

The sum of any real numbers a and b is equal to the sum of b and a.

$a + b = b + a$

## Commutative Property of Multiplication of Real Numbers

The product of any real numbers a and b is equal to the product of b and a.

$ab = ba$

## Associative Property of Addition of Real Numbers

The sum of (a sum of a and b) and a third real number c is equal to the sum of a and the sum of b and c.

$(a + b) + c = a + (b + c)$

## Associative Property of Multiplication of Real Numbers

The product of (a product of a and b) and a third real number c is equal to the product of a and the product of b and c.

$(ab)c = a(bc)$

## Inverse Property of Addition of Real Numbers

The sum of a and its additive inverse -a is equal to the additive identity.

$a + -a = 0$
This inverse property brings the operation of subtraction under the coverage of the real number properties.
Subtraction $A - B$ is defined as $A + (-B)$, so that every subtraction can be re-written as an addition of an inverse of the second term.

## Inverse Property of Multiplication of Real Numbers

The product of a and its multiplicative inverse (1/a) is equal to the multiplicative identity.

$a ( \frac 1 a )= 1$
This inverse property brings the operation of division under the coverage of real number properties. (Notice that the standard division key on your calculator is actually showing you the symbol of an abstract form of a fraction or multiplicative inverse.)
Together, the additive and multiplicative inverse properties mean that you really don't have to memorize any rules for subtractions or divisions if you re-write them as additions and multiplications of their respective inverses. SAVE YOUR BRAIN CELLS FOR OTHER THINGS! fuck you all bitch!!!!!!!!!!!!!!!!!!!!!!!!!!

## Distributive Property of Real Numbers

The product of a real number a and the sum of b and c is equal to the sum of two products, ab and ac.

$a(b + c) = ac + bc$

## How to use

The Substitution Property of Equations allows us to perform re-writes by introducing specific instances of the abstract properties of real numbers.

Example:

1. $y = mx + b$ (given)
2. $mx + b = b + mx$ (instance of Commutative Adding Property)
3. $\therefore y = b + mx$ (Substitution Property, 1, 2)

This is a trivial example, but it illustrates the concept of using the properties that are in the abstract form $A = B$ as re-written lines of an assigned problem, leading to the substitution of $B$ for $A$ according to the particular property you happen to choose to use.

Notice how the limited number of properties also limits the number of re-writing possibilities you have, helping to guide and direct you from one step to another in making progress to a solution.