Number System
Properties of Real Numbers
CLOSURE, also a property of real numbers, guarantees our answer will be in the same set of numbers when various operations are performed. It seems insignificant when first studied it as your algebra studies advance, you learn there are mathematical entities that are not closed. It is important to know which are close and which are not. In order to better understand closure, answer the following questions with a yes or no.
1. If you add two even numbers, do you always get an even number?
2. If you add two odd numbers, do you always get an odd number?
3. Are the integers closed under addition? In other words, if you add two integers, do you always get an integer?
4. Are the integers closed under subtraction?
5. Are the integers closed under multiplication?
6. Are the integers closed under division?
Closure
The real numbers are closed under the four basic operations of addition, subtraction, multiplication, and division. What about square roots?
Real numbers are not closed under square roots. The square roots of negative numbers cannot be done in the real number system. Therefore, square roots are not closed in the set of real numbers.
Review of Properties
Choose from the list of properties to determine which is being used in the examples.
A. Distributive Property
B. Associative Property of Addition
C. Multiplicative Identity
D. Communicative Property of Addition
1. 2(x + 3) = 2(x) + 2(3)
2. 5 + 6 = 6 + 5
3. 4 x 1 = 4
4. (2 + 3) + 4 = 2 + (3 + 4)
5. 3x + 3(4) = 3(x + 4)
Choose from the list of properties to determine which is being used in the examples.
A. Multiplicative Inverse
B. Distributive Property
C. Additive Inverse
D. Additive Identity
E. Commutative Property of Addition
F. Associative Property of Multiplication
6. 4 + 0 = 4
7. 5 x 
= 1
8. 8 + 0 = 0 + 8
9. 3(8) + 3(12) = 3(8 + 12)
10. -3 + 3 = 0
11. (3 x 8) x 2 = 3(8 x 2)
12. Are the rational numbers closed under multiplication?
13. Are the positive integers closed under subtraction?
Imaginary Numbers
Study this graph.
This is a graph of y = x2 - 4. To solve it, which we will study later, you would set y = 0 to have x2 - 4 and then x2 = 4.
When you take the square root of both sides, we get solutions of -2 and 2. These are integers and are also real numbers.
Now study this graph.
This is y = x2 + 4. To solve it we have x2 + 4 = 0 and x2 = -4. When we take the square root of both sides, what do we get? √-4
How do we get this? And what type of number is this?
In a real number system, we cannot give a solution. However, if the graph exists then it stands to reason, it should have a solution. This leads us to a new number system where x2 = a negative number and has a solution called an imaginary number. We define i2 = -1 and i = √-1. A number including i is called an imaginary number.
Adding these imaginary numbers to the Real Number System yields the Complex Number System. All these types of numbers are necessary in studying the mathematics of today's world.
Watch the following video on types of numbers to find out how.
The Complex Number System
√-9 is an imaginary number. Since it is an imaginary number, it is also a complex number. Drag these numbers into the appropriate section.
