Algebra
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Unit 1: Foundations for Algebra
Lesson 1.3: Real Numbers & The Number Line
How Do We Represent All the Numbers That Exist?
Also: How Do We Categorize Numbers?
Also: How Do We Categorize Numbers?
To represent all numbers that exist, we use a number line.
On the number line, you can easily see there are only two directions. The further you go to the right, the numbers get larger.
Let's look at how we classify numbers. Let start with simplest first, then get more and more complex.
Let's look at how we classify numbers. Let start with simplest first, then get more and more complex.
The Set of Natural Numbers
{1, 2, 3, 4, 5 ...}
These are also called Counting Numbers.
We have always counted things. Imagine a man in the ancient past, on the shore counting dolphins:
1 dolphin, 2 dolphins, 3 dolphins...
What about zero?
Some people include 0 in the set of natural numbers. (What if there were no dolphins that day? Seems natural to me that they would take a day off) We call the set [0, 1, 2, 3, 4...} the set of Whole Numbers.
Some people include 0 in the set of natural numbers. (What if there were no dolphins that day? Seems natural to me that they would take a day off) We call the set [0, 1, 2, 3, 4...} the set of Whole Numbers.
The Set of Integers
{...-3, -2, -1, 0, 1, 2, 3...}
The set of Integers includes both positive and negative numbers.
The set of Integers includes both positive and negative numbers.
The Set of Rational Numbers
What about all the numbers in between the integers?
Let's take any two non-zero integers and divide them. What results is the Set of Rational Numbers.
If we take any numbers a and b and write them as a ratio:
Let's take any two non-zero integers and divide them. What results is the Set of Rational Numbers.
If we take any numbers a and b and write them as a ratio:
We have the idea of a rational number. When we divide, we are taking equal parts of a into b portions. Sometimes it divides evenly. More often it does not. What results is a rational number.
Rational numbers either terminate (stop) or repeat infinitely. One divided by four is equivalent to exactly 0.25 (a terminating decimal). One divided by three is exactly 0.33333333333333....... Threes into infinity.
Rational numbers either terminate (stop) or repeat infinitely. One divided by four is equivalent to exactly 0.25 (a terminating decimal). One divided by three is exactly 0.33333333333333....... Threes into infinity.
The Set of Real Numbers
Rational Numbers either terminate or repeat. There are Irrational numbers that neither terminate nor repeat.
Rational Numbers can be written as a ratio, Irrational Numbers cannot be written as a ratio of two integers.
The Set of Real Numbers contains both rational numbers and irrational numbers.
Every real number that exists can be written on a simple number line.
One irrational number is π, the ratio of Circumference of a circle to the diameter of a circle.
Rational Numbers can be written as a ratio, Irrational Numbers cannot be written as a ratio of two integers.
The Set of Real Numbers contains both rational numbers and irrational numbers.
Every real number that exists can be written on a simple number line.
One irrational number is π, the ratio of Circumference of a circle to the diameter of a circle.
credit: mathisfun.com
More examples of irrational numbers are imperfect squares.