Real Numbers, Number Operations and the Number Line
Number Lines are an essential part of understanding Real Numbers. We started talking about real numbers and number operations in Algebra 1, this takes up where that left off. Real numbers include all the Whole Numbers 0, 1, 2, 3, …, The Integers …-3, -2, -1, 0, 1, 2, … , all the Rational Numbers …-½, 0, ¼, 1/3, ½, 1, …, and Irrational Numbers like … i, e, and √2, ….
If you're curious, you can click on Common Core to get the state standards for Numbers and Operations.
Number Operations with a Number Line
When talking about Real Numbers and Number Operations, it is common to refer to a number line. The numbers increase from left to right with the point labeled 0 being the origin. Drawing a point is called graphing the number or plotting the point.
Properties of Real Numbers
|
OF ADDITION |
OF MULTIPLICATION | |
|
Commutative Property |
a + b = b + a |
a * b = b * a |
|
Associative Property |
a + (b + c) = (a + b) + c |
a * ( b * c) = (a * b) * c |
|
Identity Property |
a + 0 = a |
a * 1 = a |
|
Inverse Property |
a + (-a) = 0 |
a * (1/a) = 1 |
|
Distributive Property |
a * (b + c) = a * b + a * c |
|
Zero Property |
a * 0 = 0 |
If you find a good way to remember these - use it. Here's what I do.
With the Commutative Property I think in terms of commuting to a job, or traveling. You move from one place to another. Notice how on both properties the numbers/letters (this is algebra) move? They relocate or Commute.l
On the Associative Property I think of how some people associate with others. If you have three people in a room, two of them might talk for a while, and then they may choose to associate with others. They didn't have to actually move - they just change their grouping.
The Identity Property asks the question - What can we do to this number that won't cause it to change? We want you to keep your original identity.
The Inverse Property is closely associated with the Identity Property. And we will be using the inverse of numbers in a lot of equations. What can I do to this number that will leave me with the identity. The Identity for addition is 0, and for multiplication it's 1.
The Distributive Property, again just as it sounds. If you distribute pamphlets, you hand one to everybody. If I'm multiplying by a, then each thing in the parenthesis gets multiplied by a.
And the Zero Property seems a bit obvious. Everything you multiply by 0 becomes 0.