Numbers: Types of Numbers

Classification of numbers:

Every number is a real number. Every number can be represented on a number line. So when we talk about the classification of numbers, we in actual talk about the Classification of real numbers:

1. Zero (0): 

It is classified as the number which represents in the case no count, no value and when we say we have none or nothing. But it is a very important value which is helpful in calculations and numerical in every field of education. It is shown in the middle of number line. Zero is also known as Neutral Integer or Neutral Number.

2. Positive and Negative Numbers: 

The numbers greater than the Zero are called Positive numbers. The numbers smaller than the Zero are called Negative Numbers. 

Positive numbers are represented in the right side of the Zero on number line. (4, 8, 9, etc)

Negative numbers are represented in the left side of the Zero on number line. (-4, -8, -9, etc)

Whole Number is defined as the combination of Positive Integers and Neutral Integer. Or Positive Integers including Zero is known as Whole Numbers.

(Figure: Number Line)

3. Rational and Irrational Numbers: 

If a number can be written in the form of p/q where p and q are prime numbers and q cannot be equal to zero, (q ≠ 0), Then number is said to be a rational number. All Integers are the rational numbers.

All terminating or recurring fractions are rational numbers.

all Non terminating but repeating fractions are rational numbers

Note: However all non terminating non repeating fractions are irrational numbers. 

For Example

p / q form of rational numbers:                            22/7, 3/4, etc

Integers as rational numbers:                             4 = 4/1= p/q where q=1≠ 0, etc

Terminating rational numbers:                            2.575, 1.25, 11.2375778

Square roots of Perfect squares:                       √4 = 2 = 2/1, etc

Cubic roots of Perfect Cubes:                            ∛27 = 3 = 3/1, etc

Non terminating repeating rational numbers: 1.333........, 1.42857142857142......, etc.

Repeating Fractions can also be represented as follows:

If a number can not be written in p / q form, they are the root expressions with results fractions or they are non terminating non repeating fractions then they are irrational numbers.

For Example

Root expressions:                                     ∛15, √5, etc

Non Terminating Non repeating fractions: 2.6457513110645905905016157536393... ... ...

Numbers are further categorized as Integers and Fractions.

Integers: An integer is a number that can be written without a non-zero fractional component. 

For Example: 7, 43,                               0,                              −17, −2048

                          ↓                                   ↓                                       ↓

               Positive Integers          Neutral Integer               Negative Integers

                                         ↘      ↙

                                             ⊕

                                              ↓ 

                                    Whole Numbers

Whole Number is defined as the combination of Positive Integers and Neutral Integer. Or Positive Integers including Zero is known as Whole Numbers.

Fractions: A fraction is a number that can be written with a non-zero fractional component. 

For Example: 2.355, 87.33333, - 20.21445, etc

Types of Integers:

(i) Prime Numbers: The integers which are only divisible by 1 along with itself, is known as prime numbers. So there are two factors of prime numbers always: 1 and number itself.

The smallest prime number is 2.

Examples of are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.

2 is the only even prime number.

Normally above 3, prime numbers normally have a formula 6k - 1 and 6k +1 with some exceptions.

Formula	k=1	k=2	k=3	k=4	k=5	k=6	k=7	k=8
6k - 1	5 ⦻	11	17	23	29	35 ⦻	41	47
6k +1	7	13	19	25 ⦻	31	37	42	49 ⦻
Formula	k=9	k=10	k=11	k=12	k=13	k=14	k=15	k=16
6k - 1	53	59	65 ⦻	71	77	83	89	95 ⦻
6k + 1	55 ⦻	61	67	73	79	85 ⦻	91	97

Note: 1 is not a prime number because it has only one factor i.e. 1.

(ii) Composite Numbers: The integers which divisible by at least one factor along with 1 and itself as its factors, is known as composite numbers.

Smallest Composite number is 4. Its factors are 1, 2, 4.

Examples are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20

(iii) Co-Prime or Relative Prime Numbers: Two integers (either Prime or Composite) are known as Relative Prime or Co-Prime Numbers if both have no any common factor except 1.

For Examples: 15 = 1× 3 × 5

                        22 = 1 × 2 × 11

We can see that both the numbers are composite numbers but they do not have any common factor along with 1. So 15 and 22 are relative prime or co-prime of each other.

Similarly, (8, 25), (10, 27), etc

(iv) Even Numbers: The integers which are completely divisible by 2 are called even numbers. An even number has 0, 2, 4, 6, 8 at its unity place or first place.

For Example: 2340, 12, 34, 56, 35798, etc

                             ↓    ↓    ↓    ↓          ↓

(The Unit place:   0,    2,   4,  6,        8)

(v) Odd Numbers: The integers which are not divisible by 2 are called odd numbers. An odd number has 1, 3, 5, 7, 9 at its unit place or first place.

For Example: 4421, 4313, 22565, 987657, 123459

                             ↓        ↓          ↓            ↓             ↓

(The Unit place:   1,        3,        5,           7,            9)

(vi) Perfect Number: An integer is a perfect number if sum of its all factors excluding itself is equal to the number itself.

For Example: 6 has factors 1, 2, 3 and 6.

                       Excluding 6 as it is the factor equal to number itself

                       1 + 2 + 3 = 6

                       We can see that sum of all factors of 6 excluding itself is equal to 6. So 6 is a perfect number.

Other example is 28 (Factors: 1, 2, 4, 7, 14, 21)

                             1 + 2 + 4 + 7 + 14 = 28, So 28 is a perfect number.

Other examples are 496, 8128, etc.

Multiples and Factors: If A number is completely divisible by an other number, then first number is called a multiple of second number and second number is called factor of first number. Take Two numbers a and b Case 1: Let a = 14 and b = 7              We can see that a is completely divisible by b. So a is multiple while b is a factor.              So 14 is multiple of 7. Similarly, 7 is a factor of 14. Case 2: Let a = 2000 and b=10              We can see that 2000 is multiple while 10 is a factor. Trick1: If number comes in a table of other number then first number is multiple and second number is factor. For Example: 14 comes in table of 7 as 7 two za 14  or 7 2s 14.                       So 14 is multiple of 7. Similarly, 7 is a factor of 14.                       2000 comes in table of 10 as 10 two hundreds za 2000  or 10 200s 2000.                       So 2000 is multiple of 10. Similarly, 10 is a factor of 2000.

Exercise for Practice

1. Which of the following is a perfect number?

(a) 8,    (b) 18,     (c) 28,    (d) 38

2. which of the following is the set of relative prime numbers?

(a) 121, 99,   (b) 141,235,   (c) 371, 203,      (d) 309, 781

3. Count prime numbers between 1 and 100 including both the numbers.

(a) 26,   (b) 25,     (c) 24,    (d) 23

4. Which of the following is not an irrational number?

(a) ∛8,     (b) ∜4,    (c) √3,     (d) 2.645751311064590590

5. What will be the a/b form of rational number 0.333?

(a) 1/3,     (b) 333/1000,      (c) Both of (a) and (b),      (d) None of These

Answer: 1 - c,     2 - d,   3 - b,      4 - a,       5 - b

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