Properties of Real Numbers

Learning Objectives

Use properties of real numbers to simplify expressions

A General Note: Properties of Real Numbers

The following properties hold for real numbers a, b, and c.

	Addition	Multiplication
Commutative Property	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative Property	$a+\left(b+c\right)=\left(a+b\right)+c$	$a\left(bc\right)=\left(ab\right)c$
Distributive Property	$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$
Identity Property	There exists a unique real number called the additive identity, 0, such that, for any real number a $a+0=a$	There exists a unique real number called the multiplicative identity, 1, such that, for any real number a $a\cdot 1=a$
Inverse Property	Every real number a has an additive inverse, or opposite, denoted $–a$ , such that $a+\left(-a\right)=0$	Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted $\Large\frac{1}{a}$ , such that $a\cdot \left(\Large\frac{1}{a}\normalsize\right)=1$

These properties allow us to simplify expressions. We simplify an expression by removing grouping symbols and combining like terms. Like terms have exactly the same variable factors. For example, $3x$ and $5x$ are like terms, because they each have exactly one factor of $x$ . On the other hand, $3x^2$ and $5x$ are not like terms because $3x^2$ has two factors of $x$ while $5x$ has just one factor of $x$ . The constant factor in a term is called its coefficient. The coefficient of $3x$ is $3$ . The coefficient of $5x$ is $5$ . The distributive property lets us combine like terms by adding their coefficients.

$3x+5x=(3+5)x=8x$

Example

Simplify each expression.

$3x+5+4x-1$
$2x+3(5x-2)$

Show Answer

$3 x + 5 + 4 x - 1$
- $=3x+5+4x+(-1)$ (Subtracting $1$ is the same as adding $-1$ )
- $=3x+4x+5+(-1)$ (Commutative Property of Addition)
- $=(3+4)x+4$ (Distributive Property; add $5$ and $-1$ )
- $=7x+4$
To simplify the expression $2 x + 3 (5 x - 2)$ we first remove the grouping symbol using the distributive property.
- $2x+3(5x-2)$
- $=2x+15x-6$
- $=(2+15)x-6$
- $=17x-6$

In the following video you will be shown how to combine like terms using the idea of the distributive property.

Example

Combine like terns:

$3x^2-5x-2+x^2+7x-3$

Show Answer

The like terms in this expression are:

$3x^2$ and $x^2$

$-5x$ and $7x$

$2$ and $-3$

We can use the commutative property to rearrange the terms so that like terms are next to each other. Since subtraction is the same as adding a negative, we can move terms we are subtracting as long as we keep the sign with the term. If a term does not have a coefficient, the multiplication identity property tells us we can give it a coefficient $1$ .

$3x^2-5x+2+x^2+7x-3$

$=3x^2+1x^2-5x+7x+2-3$

Combine like terms to obtain the simplified expression:

$4x^2+2x-1$

In the video that follows, you will be shown another example of combining like terms.

Module 6: Normal Distribution

Learning Objectives

A General Note: Properties of Real Numbers

Example

Example

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