Elementary Algebra Notes
Section 1.7
Page 1 of 2
Section 1.7: Properties of Real Numbers
Big Idea: There are five very important properties of real numbers that we must recognize in order to help us
evaluate algebraic expressions later on. They are the commutative, associative, identity, inverse, and distributive
properties.
Commutative Properties: You can add or multiply two numbers in any order. Algebraic way of saying this:
ab  ba
ab  ba
Associative Properties: If you have to add three numbers, you can add the first two first and then add the third
one, or you can add the last two first and then add the first one. The same applies for multiplication. Algebraic
way of saying this:
 a  b   c  a  b  c 
 ab  c  a  bc 
Identity Properties: There are special numbers that when added to or multiplied by any real number leaves the
original number unchanged. These special numbers are called the identity elements and are 0 (for addition) and
1 (for multiplication).
a0  a
a 1  a
The identity property is the key behind how we simplify and add fractions.
Inverse Properties: Every number has a special number called its inverse such that when the inverse is added
to or multiplied by that number, the answer is the identity element. The additive inverse is the negative of a
number, while the multiplicative inverse is the reciprocal of a number.
a   a   0
a
1
1
a
Distributive Property: If you have to multiply a sum of numbers by a factor, you can instead sum up the
products of each addend with the factor. Algebraic way of saying this:
a  b  c   ab  ac
 b  c  a  ba  ca
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 1.7
Page 2 of 2
Simplifying Algebraic Expressions by Removing Parentheses Using the Distributive Property:
Oftentimes, removing parentheses results in a simplified algebraic expression that involves one less
calculation...
Practice:
1. Simplify the algebraic expression 4  3  7  using the distributive property.
2. Simplify the algebraic expression 2  3m  4  using the distributive property.
3. Simplify the algebraic expression 6  x  y  z  using the distributive property.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.