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High School Mathematics - 2
1.1 Type of Numbers: Natural, Whole, Integers, Rational, Irrational & Real

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
Example:
2 can be expressed in the form of p/q as 2/1
-13/9 = -1.444.......
8-2 = 0.015625
(16)/3 = 4/3 = 1.333...
4 = 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.
Example: 2, 3, 7, 8
square root of 2 = 2 = 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
2 = 1. 41421356...... irrational, real
-81 = -9 integer, rational, real
-9/3 rational, real
25 = 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.
Q 1: Which of the following is an integer?
-10
all of these
0
7

Q 2: An integer is a rational number.
False
True

Q 3: Which of the following is not a natural number?
9
none of these
345
-4 0

Q 4: 16
negative
irrational
rational

Q 5: If Q1 is set of irrational numbers. What is Q1intersection of Q?
Q
null
none of these
Q1

Q 6: Q + Q1 = ____
Rational numbers Q
Real numbers R
Irrational numbers Q1

Q 7: Identify the Whole Number.
sqrt(8)
8/4
5/3
7.13

Q 8: Identify the Irrational Number.
pi
3.14
105/21
sqrt(81)

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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