1. Math humor:
When 2/0 asked it he could join the number line, he was told, “Get real!”

2. Biologists classify animals based on shared characteristics
Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko.
You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.
Animal
Reptile
Lizard
Gecko

3. The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.
Irrational numbers
Rational numbers
Real Numbers
Integers
Whole
numbers

4. Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.
4 5 23
3 = 3.8
= 0.6
1.44 = 1.2

5. Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

6. 5 is a whole number that is not a perfect square.
Examples:
Write all classifications that apply to each number.
1A. 5
5 is a whole number that is not a perfect square.
irrational, real
1B.
–12.75
–12.75 is a terminating decimal.
rational, real
16 2
= = 2
4 2
16 2
1C.
whole, integer, rational, real

7. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

8. State if each number is rational, irrational, or not a real number.
Examples:
State if each number is rational, irrational, or not a real number.
2A. 21
irrational
4 0
2C.
not a real number
0 3
0 3
= 0
2B.
rational

9. The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.

10. Find a real number between 3 and 3 . 3 5 2 5
Example 3:
Find a real number between and
3 5
2 5
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
2 5
÷ 2
3 5
5 5
= ÷ 2
1 2
= 7 ÷ 2 = 3
A real number between and is 3 .
3 5
2 5
1 2
Check: Use a graph. 3
1 5
2 5 4
3 5
4 5 3
1 2