Number System Basics | Hamaraguru

An Introduction to Numbers
A Number is considered as which has arithmetical value, expressed as a word or a symbol to represent a particular quantity. It is used for counting and in calculations. The numbers we use are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

A combination of the above numbers is used to represent quantities greater than 9. This combination is called a numeral, and each number in the numeral is called a digit. For example, 754 is a numeral and 7, 5, and 4 are digits in the numeral.

Place Value System:
Consider the numeral: 3658
The place value of the numbers increases from right to left by multiples of 10:
The place value of 8 = 8 (units place)
The place value of 5 = 5 × 10 (tens place)
The place value of 6 = 6 × 100 (hundreds place)
The place value of 3 = 3 × 1000 (thousands place)
However, the face value of each number remains the same; that is, the actual counting value of the number.

The following is a place value chart that shows the Indian as well as the international place value system:

Classification of Numbers:

Depending on how they are used, numbers are classified into the following types:

1. Natural numbers: The set of natural numbers includes all the counting numbers, starting from 1 to infinity. The symbol N is used for representing the set of natural numbers.

N = {1, 2, 3, 4, ...............}

2. Whole numbers: The set of whole numbers includes 0 and all natural numbers. The symbol W is used to represent the set of whole numbers.

W = {0, 1, 2, 3, ...............}

Note: All natural numbers are also whole numbers. That is, the set of natural numbers is a subset of the set of whole numbers: N ⊂ W

3. Integers: The set of integers includes 0, positive numbers from 1 to infinity, and negative numbers from -1 to negative infinity. The symbol I or Z is used to represent the set of integers (in this book, we have used the symbol I).

I = {............... -3, -2, -1, 0, 1, 2, 3, ...............}

Or

Z = {   -5,-4. -3, -2, -1, 0, 1, 2, 3,4,5   }

Note 1: All whole numbers are also Integers. That is, the set of whole numbers is a subset of the set of integers: W ⊂ I

Note 2: It is convenient to represent integers on the number line as follows:

4. Rational numbers: The set of numbers that can be represented in the form p/q such that p and q are integers and q is not 0. The symbol Q is used for representing the set of rational numbers.

Examples: 2/3, 3.5, 4, etc.

Note: All integers are rational numbers. That is, the set of integers is a subset of the set of rational numbers: I ⊂ Q

5. Irrational numbers: The set of numbers that are not rational are said to be irrational numbers. That is, these numbers cannot be expressed in the form p/q such that p and q are integers and q is not 0.

Examples: √2, √3, √7, √11, π, etc.

(π is an irrational number. The value 22/7 which is a rational number is only an approximate value of π and is used to simplify calculations.)

Note: Irrational numbers can be represented on the number line.

6. Real numbers: The set of real numbers include all numbers that can be represented on the number line. The symbol R is used for representing the set of real numbers. All rational numbers along with irrational numbers are real numbers.

Both rational numbers, as well as irrational numbers, are subsets of the set of real numbers.

Q ⊂ R; and {Irrational numbers} ⊂ R.

Note: All rational numbers, as well as irrational numbers, are real numbers.

7. Imaginary numbers: These numbers are square roots of negative integers. The symbol i is used to representing the square root of i = 1.

All imaginary numbers are represented as multiples of i.

For example, − 2 = 2 × − 1 = 2 × − 1 = 2 i.

More examples of imaginary numbers are: 2 7

2i, -3i ,3i/17  etc.

8. Complex numbers: These numbers are a combination of real and imaginary numbers. They are generally written in the form a + bj; where a and b are real numbers, and i = 1. The symbol C is used to represent the set of complex numbers.

Examples: 2 + 3i, 7 – 9i, etc.

The set of real numbers is a subset of the set of complex numbers:  R ⊂ C.

Note: Complex and imaginary numbers are used in algebra and higher level and theoretical mathematics.

The simple arithmetic that we are going to deal with in numerical ability is limited to the set of real numbers.

There are other classifications of numbers that are used in regular calculations:

Consecutive Numbers are numbers that occur one after the other in ascending order.

For example:

1. 65, 66, 67 are consecutive natural numbers.

2. 0, 1, 2, 3, 4, 5 are consecutive whole numbers.

3. -32, -31, -30, -29, -28, -27, -26 are consecutive integers.

4. 4, -2 , 0, 2, 4 are consecutive even numbers including 0 also.

5. -3,-1, 1, 3, 5, 7, are consecutive odd numbers.

6. 2, 3, 5, 7, 11, 13, are consecutive prime numbers.

Even and Odd Numbers:

Even numbers are multiples of 2. In other words, all numbers that end with 0, 2, 4, 6, and 8 are even numbers.

For example, 2, 4, 6, 8, 10, and 12 are even numbers.

Odd numbers are not divisible by 2. In other words, all numbers that are not multiples of 2 are odd numbers.

For example, 1, 3, 5, 7, and 9 are odd numbers.

Note1: All numbers ending with 1, 3, 5, 7, and 9 are odd numbers.

Note2: In a sequence of consecutive numbers, even and odd numbers appear alternately.

For example, consider the following sequence of consecutive numbers:

Remember:

1. Odd no. + odd no. = even no.

2. Even no. + even no. = even no.

3. Odd no. + even no. = odd no.

4. Odd no. – odd no. = even no.

5. Even no. – even no. = even no.

6. Odd no. – even no. = odd no.; even no. – odd no. = odd no.

7. Odd no. × odd no. = odd no.

8. Even no. × even no. = even no.

9. Odd no. × even no. = even no.

10. Odd no ÷ odd no = odd no (if the number is divisible)

11. Odd numbers are not divisible by even numbers.

12. Even no ÷ odd no = even no (if the number is divisible)

Factors and Multiples:

If a number can be exactly divisible by another number, then the remainder of the division is 0.

For example: 21÷ 3 = 7

This means that 21 can be divided by 3 exactly 7 times and the remainder after the division is 0.

Therefore, 21 is said to be multiple of 7.

Now, 7 × 3 = 21

That is, 7 and 3 yield 21 when multiplied.

Hence, 7 and 3 are said to be the factors of 21.

Also 21 = 1 × 21

Therefore, 1 and 21 are also factors of 21.

Note:

• Every number is a factor of itself

• 1 is a factor of every number

Divisibility Rules:

Divisibility by 2

The rule for being divisible: Any even number

Example:

1. 348 is an even number because it ends with an even number. Therefore, it is divisible by 2.

2. 875 is an odd number because it ends with an odd number. Therefore, it is not divisible by 2.

Divisibility by 3

The rule for being divisible: The sum of the digits is a multiple of 3.

Example:

1. 987: 9 + 8 + 7 = 24; 2 + 4 = 6; 6 is a multiple of 3. Therefore, 987 is divisible by 3.

2. 253: 2 + 5 + 3 = 10; 10 is not a multiple of 3. Therefore, 253 is not divisible by 3.

Divisibility by 4

The rule for being divisible: The number is even, and the number formed by the last two digits is a multiple of 4.

Example:

1. 4356: The last two digits are 56, which is a multiple of 4. Therefore, 4356 is divisible by 4.

2. 5678: The last two digits are 78, which is not a multiple of 4. Thus, 5678 is not divisible by 4.

Divisibility by 5

The rule for being divisible: The number ends with 0 or 5.

Example:

1. 7860: Ends with a 0. Therefore, 7860 is divisible by 5.

2. 6785: Ends with a 5. Therefore, 6785 is divisible by 5.

3. 3968: Does not end with 5 or 0. Therefore, 3968 is not divisible by 5.

Divisibility by 6

The rule for being divisible: The number is divisible by 2 and 3. (since 6 = 2 × 3)

Example:

1. 126: It is even; and 1 + 2 + 6 = 9, which is a multiple of 3. Therefore, 126 is divisible by 6.

2. 519: It is odd. Therefore, 519 is not divisible by 6.

3. 728: It is even; but 7 + 2 + 8 = 17, which is not a multiple of 3. Therefore, 728 is not divisible by 6.

Divisibility by 7

The rule for being divisible: Double the last digit, subtract this number from the remaining number. If the answer is either 0 or a multiple of 7, the given number is divisible by 7.

Example:

1. 1029: Last digit is 9.

Double of 9 is 18.

Remaining number is 102.

102 − 18 = 84.

Since 84 is a multiple of 7, 1029 is divisible by 7.

2. 681: the last digit is 1.

Double of 1 is 2.

The remaining number is 68.

68 − 2 = 66.

66 is not divisible by 7.

Hence, 681 is not divisible by 7.

Divisibility by 8

The rule for being divisible: The number is even, and the number formed by the last three digits is a multiple of 8.

Example:

1. 7064: The last three digits form the number 064 or 64, which is a multiple of 8. Therefore, 7064 is divisible by 8.

2. 4589 is odd. Therefore, it is not divisible by 8.

3. 1562 is even, but 562 is not a multiple of 8. Therefore, it is not divisible by 8.

Divisibility by 9

The rule for being divisible: The sum of the digits is a multiple of 9.

Example:

1. 228: 2 + 2 + 8 = 12, which is not a multiple of 9. Therefore, 228 is not divisible by 9.

2. 5616: 5 + 6 + 1 + 6 = 18, which is a multiple of 9. Therefore, 5616 is divisible by 9.

Divisibility by 10

The rule for being divisible: The number ends with 0.

Example:

1. 560: Ends with 0. Therefore, 560 is divisible by 10.

2. 564: Does not end with 0. Therefore, 564 is not divisible by 10.

Divisibility by 11

The rule for being divisible: Add alternate digits and find their difference. If this difference is either 0 or a multiple of 11, the given number is divisible by 11.

Example:

1. 14641: The sum of alternate digits is (1 + 6 + 1 = 8) and (4 + 4 = 8); difference = 8 - 8 = 0. Therefore, 14641 is divisible by 11.

2. 1408: The sum of alternate digits is (1 + 0 = 1) and (4 + 8 = 12); difference = 12 − 1 = 11, which is a multiple of 11. Therefore, 1408 is divisible by 11.

3. 12321: The sum of alternate digits is (1 + 3 + 1 = 5) and (2 + 2 = 4); difference = 5 − 4 = 1, which is neither 0 nor a multiple of 11. Therefore, 12321 is not divisible by 11.

Divisibility by 12

The rule for being divisible: The number is divisible by 3 and 4. (since 12 = 3 × 4).

Example:

1. 1284: 1 + 2 + 8 + 4 = 24, which is a multiple of 3, and 84 is a multiple of 4. Therefore, 1284 is divisible by 3 and 4 and hence 12.

2. 2498: 98 is not a multiple of 4. Therefore, 2498 is not divisible by 12.

3. 9851 is odd and hence not divisible by 12.

Divisibility by 13

The rule for being divisible: Method 1

Find four times the last digit, Add this number to the remaining number.

Continue this process until you get a two-digit number.

If this number is a multiple of 13, then the given number is divisible by 13.

Example:

1. 9139:

a. Four times last digit = 36, remaining number = 913; 913 + 36 = 949

b. Four times last digit = 36, remaining number = 94; 94 + 36 = 130

130 is a multiple of 13, therefore, 9139 is divisible by 13.

2. 5075:

a. Four times last digit = 20; remaining number = 507; 507 + 20 = 527

b. Four times last digit = 28; remaining number = 52; 52 + 28 = 80. 80 is not a multiple of 13, therefore, 5075 is not divisible by 13.

The rule for being divisible: Method 2

Find nine times the last digit.

Subtract this number from the remaining number.

Continue this process till you arrive at either 0 or a two-digit number.

If the result is either 0 or a multiple of 13, then the given number is divisible by 13.

Example:

1. 9139:

a. Nine times last digit = 81; remaining number = 913; 913 − 81 = 832

b. Nine times last digit = 18, remaining number = 83; 83 − 18 = 65

c. 65 is a multiple of 13, therefore, 9139 is divisible by 13.

2. 5075:

a. Nine times last digit = 45, remaining number = 507; 507 − 45 = 462

b. Nine times last digit = 18, remaining number = 46; 46 − 18 = 28

c. 28 is a not a multiple of 13, therefore, 5075 is not divisible by 13.

Divisibility by 14

The rule for being divisible: The number is divisible by 2 and 7. (Since 14 = 2 × 7).

Example:

1. 1176 is divisible by 2 and 7. Therefore, it is divisible by 14.

2. 1261 is odd and hence not divisible by 2. Therefore, it is not divisible by 14.

Divisibility by 15

The rule for being divisible: The number is divisible by 3 and 5. (Since 15 = 3 × 5).

Example:

1. 1020: 1 + 0 + 2 + 0 = 3, which is a multiple of 3; it ends with 0, and is hence divisible by 5. Therefore, 1020 is divisible by 15.

2. 1413 does not end with 0 or 5 and. Therefore, it is not divisible by 5 and hence by 15 as well.

3. 985 ends with 5. But 9 + 8 + 5 = 22, which is not a multiple of 3. Therefore, 985 is not divisible by 15.

Divisibility by 16

The rule for being divisible :

This test can be used for large numbers.

Case 1: The thousands digit is even Check if the number formed by the last three digits is a multiple of 16. If it is divisible, then the given number is divisible by 16.

Example:

1. 1048576: The thousands digit is even. The last three digits form the number 576, which is a multiple of 16. Therefore, the given number is divisible by 16.

2. 312698: The thousands digit is even. The last three digits form the number 698, which is not a multiple of 16. Therefore, the given number is not divisible by 16.

Case 2:

The thousands digit is odd.

1. Add 8 to the last three digits.

2. If the result is exactly divisible by 16, then the given number is divisible by 16.

Example:

1. 16777216: The thousands digit is odd. The last three digits form 216. 216 + 8 = 224, which is a multiple of 16. Therefore, the given number is divisible by 16.

2. 65538: The thousands digit is odd. The last three digits form 538. 538 + 8 = 546, which is not a multiple of 16. Therefore, the given number is not divisible by 16.

Divisibility by 17

The rule for being divisible :

1. Find 5 times the last digit

2. Subtract this result from the remaining number.

3. Continue this process until you get at a two-digit number.

4. If the result is either 0 or a multiple of 17, then the given number is divisible by 17.

Example:

1. 83521:

a. Five times last digit = 5; remaining number = 8352; 8352 − 5 = 8347

b. Five times last digit = 35; remaining number = 834; 834 − 35 = 799;

c. Five times last digit = 45; remaining number = 79; 79 − 45 = 34.  34 is a multiple of 17, therefore, 83521 is divisible by 17.

2. 78615:

a. Five times last digit = 25; remaining number = 7861; 7861 − 25 = 7836

b. Five times last digit = 30; remaining number = 783; 783 − 30 = 753

c. Five times last digit = 15; remaining number = 75; 75 − 15 = 60. 60 is not a multiple of 17, therefore, 78615 is not divisible 17.

Divisibility by 18

The rule for being divisible: The number is divisible by 2 and 9. (Since 18 = 2 × 9).

Example:

1. 3672 is even and 3 + 6 + 7 + 2 = 18, which is a multiple of 9. Therefore, 3672 is divisible by 18.

2. 2465 is odd, and is hence not divisible by 18.

3. 4592 is even, but 4 + 5 + 9 + 2 = 20, which is not a multiple of 9. Therefore, 4592 is not divisible by 18.

Divisibility by 19

The rule for being divisible :

1. Find twice the last digit

2. Add this result to the remaining number

3. Continue this process unless you arrive at a two-digit number.

4. If the resulting number is a multiple of 19, then the given number is divisible by 19.

Example:

1. 6859:

a. Twice last digit = 18, remaining number = 685; 685 + 18 = 703

b. Twice last digit = 6; remaining number = 70; 70 + 6 = 76. 76 is a multiple of 19, therefore, 6859 is divisible by 19.

2. 14932:

a. Twice last digit = 4; remaining number = 1493; 1493 + 4 = 1497

b. Twice last digit = 14; remaining number = 149; 149 + 14 = 163

c. Twice last digit = 6; remaining number = 16; 16 + 6 = 22. 22 is not a multiple of 19, therefore, 14932 is not divisible by 19.

Divisibility by 20

The rule for being divisible: An even number is followed by 0.

Example:

1. 3140 ends with 0, and 314 is even. Therefore, it is divisible by 20.

2. 2170 ends with 0, but 217 is not even. Therefore, it is not divisible by 20.

3. 4789 does not end with 0. Therefore, it is not divisible by 20.

Note: This can be extended to all multiples of 10:

• Divisibility by 30: A number divisible by 3 is followed by 0.

• Divisibility by 40: A number divisible by 4 is followed by 0.

• Divisibility by 50: A number divisible by 5 is followed by 0.

• Divisibility by 60: A number divisible by 6 is followed by 0.

Remember:

• Every number is divisible by 1.

• Every number is divisible by itself (1 time).

• The product of two consecutive whole numbers is always even (example: 3 × 4 = 12).

• If any number x is divisible by another number y then x is divisible by each factor of y (example: 108 is divisible by 12; therefore, 108 is divisible by 1, 2, 3, 4, 6, and 12—factors of 12).

• If x and y are divisible by a, then (x + y) and (x − y) are also divisible by a (example: 4 and 6 are

divisible by 2. Therefore, 4 + 6 = 10 and 6 − 4 = 2 are also divisible by 2).

If n is divisible by x, then:

We can write n as the sum of two numbers such that each is divisible by x (example: 12 is divisible

by 4; 12 = 8 + 4 such that 8 and 4 are divisible by 4).

Or we can write n as the sum of two numbers such that each is not divisible by x (example: 12 is

divisible by 4; 12 = 5 + 7 such that 5 and 7 are not divisible by 4).

We cannot write n as the sum of two numbers such that one is divisible and the other is not divisible by x.

We can write n as the difference of two numbers such that each is divisible by x (example: 20 is divisible by 5; 20 = 55 − 35 such that 55 and 35 are divisible by 5).

Or we can write n as the difference of two numbers such that each is not divisible by x (example: 20 is divisible by 5; 20 = 28 − 8 such that 28 and 8 are not divisible by 5).

We cannot write n as the difference of two numbers such that one is divisible and the other is not  divisible by x.

Prime and Composite Numbers:
A prime number has only two factors – one and itself. That is, it is divisible only by 1 and itself.

For example: 2, 3, 5, 7, 11, 13, …….. are all prime numbers.

All other numbers except 1 are known as composite numbers. These numbers are divisible by at least 3 numbers.

For example 1. 4 is divisible by 1, 2, and 4. Therefore it is a composite number.
Similarly, 15 can be divided by 1, 3, 5, and 15. Therefore it is a composite number.
Note:
1. The number 1 is neither said to be a prime nor a composite.
2. The number 2 is the only number which is an even number and a prime number.

Coprime Numbers: Two numbers are said to be coprime to each other if they have only 1 as their common
factor. They are also said to be relatively prime or mutually prime.
For example:
35 and 36 have no factor in common except 1. Hence, they are coprime.
35 and 42 have 7 and 1 as common factors. Therefore, they are not coprime.

Note:
1. The HCF of two co-prime numbers is 1.
2. If x and y are co-prime, and n is divisible by each of them, then n is also divisible by
xy.
For example:
a. 1050 is divisible by 2 and 5; 2 and 5 are co-prime; hence, it is also divisible by 10.
b. 1050 is divisible by 2 and 10; 2 and 10 are not co-prime; hence, it is not divisible by 20.

Testing for a Prime Number:
1. Let the given number be n.
2. Find a whole number which is greater √n. Let this be a.
3. Check if n is divisible by any of the prime numbers < = a.
4. If it is not divisible, then n is a prime number.

For example:
1. Determine whether 57 is a prime number.
64 is the perfect square that is close to 57. Therefore 8 >√57
Prime numbers < = 8 are 2, 3, 5, 7
a. 57 is odd and hence not divisible by 2
b. 5 + 7 = 12, which is a multiple of 3
Therefore, 57 is not a prime number.

2. Is 73 a prime number?
81 is the square that is close to 73. Therefore 9 > √73
Prime numbers < = 9 are 2, 3, 5, 7
a. 73 is odd and hence not divisible by 2
b. 7 + 3 = 10, which is not a multiple of 3. Therefore, it is not divisible by 3.
c. 73 does not end with 0 or 5 and is hence not divisible by 5.
d. 73 is not a multiple of 7 and is hence not divisible by 7.
Therefore, 73 is a prime number.

Let us try the same principle with bigger numbers
3. Is 377 a prime number?
400 is the square that is close to 377. Therefore approximate square root of 400 = 20
Prime numbers < = 20 are 2, 3, 5, 7, 11, 13, 17, 19
a. 377 is odd and hence not divisible by 2
b. 3 + 7 + 7 = 17, which is not a multiple of 3. Therefore, it is not divisible by 3.
c. 377 does not end with 0 or 5 and is hence not divisible by 5.
d. Double last digit = 14; remaining number = 37; 37 − 14 = 23, which is not a multiple of 7. Therefore, it is not divisible by 7. (if the calculator is allowed, divide 377 by 7. If the answer is a whole number, it
is divisible. If the answer is a decimal it is not divisible.)
e. (3 + 7) − 7 = 3, which is neither 0 nor a multiple of 11. Therefore, it is not divisible by 11. (use a calculator if allowed)
f. 37 + (4 × 7) = 65, which is a multiple of 13. Therefore, it is divisible by 13. Therefore, 377 is not a prime number.

Prime Factorization: In prime factorization, we express a number as the product of its prime factors.
To find the prime factorization of a given number, we keep breaking down the factors of the number until all of them are prime.
For example: Find the prime factors of 42
We repeatedly break down the factors of 42 until all of them are prime
Thus, 42 = 2 × 3 × 7

Perfect Numbers:

A positive integer is called to be a perfect number if it is equal to the sum of all its factors (except the number itself).
For example, we consider the number 6. The factors of 6 are 1, 2, 3, and including 6 itself. Now all factors except itself are 1, 2, and 3;
and
sum of these factors = 1 + 2 + 3 = 6.
Therefore, 6 is a perfect number. 6 is the positive integer, which is the smallest perfect number.
Note:
• If the sum of the factors is greater than the number, it is said to be abundant.

For example:
• If the sum of the factors is smaller than the number, it is said to be deficient.