# MandysNotes

## Properties of Real Numbers

Wednesday, 12 February 2014 02:03

### Sets, Rings, and Groups of Numbers.

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Sets, Rings, and Groups of Numbers

Sunday, 21 March 2010 18:05

### The Multiplication Property of Equality

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For all real numbers A, B and for when C does NOT equal 0, the equations:

A = B and AC = BC are equivalent to one another.

Each side can be multiplied by the same NONZERO number without changing the solution set.

Sunday, 21 March 2010 18:05

### The Addition Property of Equality

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For all real numbers A, B and C, the equations:

A = B and A + C = B + C are equivalent to one another.

The same number may be added to each side of an equation without altering the solution set.

Saturday, 20 March 2010 20:14

### The Associative Property

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The Associative Property states that for any real numbers a, b and c:

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

a(bc) = (ab)c

With the Associative Property, parentheses amongst 3 terms/factors change, but the order of the terms stays the same.

Saturday, 20 March 2010 20:12

### Commutative Property

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The Commutative Property states:

For any real numbers a and b:

a + b = b + c

ab = ba

The Commutative Property is when the ORDER CHANGES but the result remains the same.

Saturday, 20 March 2010 20:05

### Identity Properties

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For any real number a,

a + 0 = 0 + a = a

a * 1 = 1 * a = a

Notes: An easy way to remember this is that the Identity Property leaves the IDENTITY of a real number unchanged. Adding 0 to any number or multiplying any number times a does not change the value of the number in any way.

EXAMPLES:

12m + m

= 12m + 1m

= (12 + 1)m

= 13m

Saturday, 20 March 2010 00:00

### Inverse Properties

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For any real number $$a \neq 0 :$$

$$a + (-a) = 0 \$$ and,

$$a \left( \frac{1}{a} \right) = 1.$$

That is, any number times its reciprocal equals 1.

The reciprocal of zero is not defined.

Saturday, 20 March 2010 20:00

### The Distributive Property, Explanation & Examples

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Very simply stated, the Distributive Property is easy to understand when seen like this:

a(b+c) = ab + ac

OR

(b+c)a = ba + ca

In more complex examples, the Distributive Property can be applied to equations like this one, though the same principles still apply:

EXAMPLES:

3(x + y)

= 3x + 3y

-2(5 + k)

= -2(5) + (-2)(k)

= -10 - 2k

4x + 8x

= (4 + 8)x

= 12x