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Natural Numbers(N):
Natural Numbers are counting numbers from 1,2,3,4,5,................
N = {1,2,3,4,5,................}
Whole Numbers (W):
Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5,...............
W = {0,1,2,3,4,5,..............}
W = 0 + N
Positive Numbers:
Positive numbers are, 1,2 ,3 ,4 ,5.................
Positive Numbers: {1, 2, 3, . . .}
Negative Numbers:
Negative numbers are, ............-3, -2, -1.
Negative integers: { . . . -3, -2, -1}
Integers (Z):
- Whole Numbers together with negative numbers.
- Integers are set containing the positive numbers, 1, 2, 3, 4, ...., and negative numbers,............-3, -2, -1, together with zero.
- Zero is neither positive nor negative, but is both.
- In other words, Integers are defined as set of whole numbers and their opposites.
- Z = {..., -3, -2, -1, 0, 1, 2, 3, .....}
Rational Numbers (Q):
- All numbers of the form , where a and b are integers (but b cannot be zero)
- Rational numbers include fractions:
* Proper Fraction: Numbers smaller than 1 eg: 1/2 or 3/4
* Improper Fraction: Numbers greater than 1 eg: 5/2
* Mixed Fraction: 2 1/2 = 5/2
- Powers and square roots may be rational numbers if their standard form is a rational number.
- In rational numbers the denominator cannot be zero
2 can be expressed in the form of p/q as 2/1
-13/9 = -1.444.......
8-2 = 0.015625
(Ö16)/3 = 4/3 = ±1.333...
Ö4 = 2 1/2 = 0. 5 ----- Rational (terminates)
2/3 = 0.6666666.......Rational (repeats)
5/11 = 0.454545......Rational (repeats)
Irrational Numbers Q1:
- Cannot be expressed as a ratio of integers.
- As decimals they never repeat or terminate (rationals always do one or the other)
- They go on for ever or infinity.
square root of 2 = Ö2 = 1. 41421356......Irrational (never repeats or terminates)
pi = p = 22/7 = 3.14159265....... Irrational (never repeats or terminates)
Real Numbers R:
- Real Numbers are every number, irrational or rational.
- Any number that you can find on the number line.
- It is a number required to label any point on the number line; or it is a number that names the distance of any point from 0.
- R = Q + Q1
- Natural Numbers are Whole Numbers, which are Integers, which are Rational Numbers, which are Real Numbers.
- Irrational Numbers are Real Numbers, but not all Real Numbers are Irrational Numbers.
Examples:
| 0.45 | rational real |
| 3.1415926535................... | irrational, real |
| 3.14159 | rational, real |
| 0 | whole, integer, rational, real |
| 5/3 | rational, real |
| 1 2/3 = 5/3 | rational, real |
| Ö2 = 1. 41421356...... | irrational, real |
| -Ö81 = -9 | integer, rational, real |
| -9/3 | rational, real |
| Ö25 = 5 | natural, whole, integer, rational, real |
| 9/3 = 3 | natural, whole, integer, rational, real |
| -3/4 | rational, real |
| p = 3.1428571... | irrational, real |
| 3.144444....... | rational, real (since it is a repeating decimal) |
Directions: Choose the correct answer. Also, write five examples of your own for each type of number.