# Number System

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The numbers like 1,2,3,… are used for counting and they are known as natural numbers. If two natural numbers are added, the resulting number will also be a natural number. However, subtraction of one natural number from another natural number may not result in another natural number, e.g. 2-6 = -4 which is not a natural number. This difficulty is overcome by the introduction of negative numbers -1, -2, -3,… and 0.

The set of natural numbers, zero and negative numbers constitute integers.

Rational numbers:

A rational number is the ratio of two integers p and q (p and q are either positive or negative) and are in the form p/q, where q is not equal to zero. Examples of natural numbers are 3/5, -7/3, 4. 4 is a rational number as 4 = 4/1.

Irrational numbers:

The numbers which are not rational are called irrational numbers. _/2 and pi cannot be expressed as ratio of 2 integers (as fractions) and hence they are irrational numbers. Their values are approximately 1.414… and 3.14159… which are non-terminating decimals.
Among irrational numbers, there are numbers which when converted to decimal form are non-terminating decimals. E.g.,

1/3 = 0.333…

1/7=0.142857142857…

In case of rational numbers, the decimal parts repeat themselves, whereas it is not the case in respect of irrational numbers.

The usual notations are:

Z defines the set of integers like -2, -1, 0,1, 2…

N defines the set of rational numbers like 2/3, -5/6, 5, …

R defines real numbers

C defines complex numbers.

Note: 0 is not taken to be a positive or negative integer, but it is a whole number (integer).

Even numbers:

Numbers that are multiples of 2 are even numbers, i.e. 2, 4, 6, 8, ….are even numbers. Numbers that are not multiples of 2 are odd numbers.

(i) In general 2n is an even number and 2n + 1 and 2n – 1 are odd numbers (where n is a whole number).

(ii) The sum or difference of 2 evem numbers is even,

e.g., 18 + 22 = 40

40 – 24 = 16

(iii) The sum of two odd numbers is even and the difference of two odd numbers is also even.

e.g., 25 + 31 = 56

45 - 29 = 16

(iv) Any power of an even number is even

e.g., 2^{4} = 16; 6^{3} = 216

(v) Any power of an odd number is odd

e.g., 3^{2} = 9; 5^{3} = 125

(vi) An even number multiplied by an even number is even

e.g., 6 x 4 = 24

(vii) An odd number multiplied by an odd number is odd

e.g., 3 x 7 = 21

(viii) An odd number multiplied by an even number is even

e.g., 7 x 6 = 42

**Define the divisibility test in number system?**

**Solution:** The test of divisibility in number system is given below:

**Test of Divisibility**

(i) **A number divisible by 2 or 5**

Any number ending in 0 or an even number is divisible by 2 e.g. 12, 256, 328, 2060. If the last digit of a number is 0 or 5, that number is divisible by 5 e.g. 150, 2025, 3175.

(ii)** A number divisible by 4 or 25**

Any number is divisible by 4 if the last two digits are divisible by 4, e.g. 132, 5276, 208. Similarly if the last two digits of a number are divisible by 25, that number is divisible by 25, e.g. 1375, 2500.

(iii) **A number divisible by 8 or 125**

For a number to be divisible by 8, its last three digits must be divisible by 8, e.g. 864, 1248, 3000. Similarly numbers ending with the last three digits divisible by 125 are divisible by 125, e.g. 4250, 12375, 12000.

(iv) **A number divisible by 16 or 625**

A number having its last four digits divisible by 16, will be divisible by 16, e.g. 31776, 28528. Similarly numbers having their last four digits divisible by 625 are divisible by 625, e.g. 83125, 125000.

(v) **A number divisible by 3 or 9**

If a number is divisible by 3, the sum of the digits is divisible by 3, e.g. 38451, 285612.
In the above numbers, sum of digits 3+8+4+5+1 = 21 and 2+8+5+6+1+2 = 24, which are all divisible by 3. Hence, the numbers are divisible by 3.

In a similar manner, a number is divisible by 9, if the sum of all its digits is divisible by 9, e.g. 1548, 653229.

In the above numbers, the sum of digits = 1+5+4+8 = 18, 6+5+3+2+2+9 = 27. Hence, the numbers are divisible by 9

(vi) **A number divisible by 11**

A number is divisible by 11 if the difference between the sum of the digits in the even places and the sum of the digits in the odd places is either 0 or a number divisible by 11, e.g. 65274, 538472.

In 65274, difference in the sum of numbers in the odd places and the sum of numbers in the even places = (6+2+4) – (5+7) = 0

In 538472, difference in the sum of numbers in the odd places and the sum of numbers in the even places = (5+8+7) – (3+4+2) = 20 – 9 = 11, a multiple of 11. Hence the two numbers are divisible by 11

**How to solve the following problem?**

A and B start on a certain journey together, A walking 5 kn an hour, B riding 8 km an hour. B stops for half an hour on the journey but finally arrives at his destination 1 hour before. What is the length of the journey?

**Solution:**

Let the length of the journey be x km.

Then the time taken by A to reach the destination = x/5 hours.

The time for which B rides = x/8 hours.

But he stops for half an hour.

Total time spent by B to reach the destination = x/8 + ½ hours.

Since B arrives 1 hour before A, we have

(x/8 + ½ + 1) = x/5

x/8 + 3/2 = x/5

Now multiplying both sides by 40 (LCM of 8, 2, and 5), we get

5x + 60 = 8x

60 = 8x 5x

3x = 60 or, x = 20

Hence, the length of journey is 20 km.

Check:

Time taken by A = 20/5 = 4 hours

Time taken by B = 20/8 + ½ = 5/2+1/2 = 3 hours