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The Number System

  1. Posted by Max in The Number System |
  2. November 25th, 2009 |
  3. Comments off

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. √2and 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., 24 = 16; 63 = 216

(v) Any power of an odd number is odd
e.g., 32 = 9; 53 = 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

Example: The sum of a two digit number and the number formed by reversing the digits is a perfect square. Find the numbers.

Given a number 29 with 2 digits, the number got by reversing the digits is 92.
29 could be written as 2×10 + 9
92 could be written as 9×10 + 2
The sum of the two numbers is 29 + 92 = 121 = 112
In general, if the digit in the ten’s place is x and the unit place is y, then the number is 10x + y.
The number on reversing becomes 10y + x.
Sum of the 2 numbers is
(10x + y) + (10y + x) = 11(x+y)
The maximum value of x or y can be 9.
Therefore, x+y= (9+9)
Implies, x+y = 18
Ii(x+y) must be a perfect square.
This is possible only when x+y = 11
The possible number of x and y are given by
(x,y) = (2,9) or (9,2)
(x,y) = (3,8) or (8,3)
(x,y) = (4,7) or (7,4)
(x,y) = (5,6) or (6,5)
Hence, the required numbers are 29,92; 38,83; 47,74; 56,65;
For example, 38 + 83 = 121 = 112
Consecutive numbers are those numbers, such that the difference of any number from the previous number is 1.
2,3,4,5,6,7,8,… are consecutive numbers
2,4,6,8,… are consecutive even numbers
3,5,7,9,… are consecutive odd numbers
Example: Convert the repeating decimal expansion 3.3333 a a rational number
Let x = 3.3333
10x = 33.33
Subtracting (1) from (2), we get
10x – x = (33.3333….-3.3333) = 33 – 3 = 30
i.e., 9x = 30
Implies, x = 30/9 = 10/3 = 3-1/3

Example: Express 2.5737373737373…. as a fraction of the form p/q
2.5737373737373… means the decimal part 73 repeats itself
Let x = 2.573737373… (i)
10x = 25.73737373… (ii)
1000x = 2573.737373… (iii)
Subtracting (ii) from (iii), we get
1000x – 10x = (2573.737373) – (2.5737373…)
= 2573 – 25 = 2348
i.e., 990x = 2548
Therefore, x = 2548/990 = 1274/495