**NUMBER SYSTEM NOTES FOR COMPETITIVE EXAMS**

Dear Aspirants, We have noted down some **short notes on number system** for the competitive exams. You all should revise below all notes before giving any exams. It will make your knowledge deeper and help in scoring good marks.

**(a)** 2 is the only even prime number.

**(b)** Number 1 is neither divisible nor prime.

**(c)** Two consecutive odd prime numbers are called prime pair.

**(d)** All natural numbers are whole, rational, integer and real.

**(e)** All whole numbers are rational Integer and real.

**(f)** All whole numbers are rational and real.

**(g)** All whole numbers, rational and irrational numbers are real.

**(h)** Whole numbers and natural numbers can never be negative.

**(i)** Natural (including Prime, Composite, even or odd) numbers and whole numbers are never negative.

**(j)** Fractions are rational.

**(k)** All prime number except 2 are odd.

**(l)** 0 is neither negative nor positive number.

**(m)** If a is any number then, if a divides zero, result will be zero. If 0 divides a, then result will be infinite or not defined or undetermined i.e.

$$\frac { a }{ 0 } =0\quad but\quad \frac { a }{ 0 } =\infty (infinite)$$

where a is real number.

**(n)** Dividing 0 by any number gives zero e.g. $$\frac { 0 }{ a } =0$$

**(o)** The place or position of a digit in a number is called its place value such as Place value of 2 in 5283 is 200.

**(p)** The real value of any digit in a certain number is called its face value. As, face value of 2 in 5283 is 2.

**(q)** The sum and the product of two rational rational number is always a rational numbers.

**(r)** The product or the sum of a rational number and irrational number is always an irrational number.

**(s)** π is an irrational number.

**(t)** There can be infinite number of rational or irrational numbers between two rational numbers or two irrational numbers.

**(u)** Decimal indication of an irrational number is infinite coming. $$as\quad -\sqrt { 3 } ,\sqrt { 2 } $$

**(v)** The square of an even and the square of an odd number is odd.

**DECIMAL**

**(w)** The decimal representation of a rational number is either finite or infinite recurring e.g. $$=\frac { 3 }{ 4 } =0.75(finite)$$

$$\frac { 11 }{ 3 } =3.666…(infinite\quad recurring)$$

**(x)** If decimal number is 0, x and 0. xy are given, then they can be expressed in the form of $$\frac { p }{ q } $$

$$As,\quad 0.x\quad =\quad \frac { x }{ 10 } \quad and\quad 0.xy\quad =\quad \frac { xy }{ 100 } $$

**(y)** If decimal recurring numbers $$0.\overline { x } \quad and\quad o.\overline { xy } \quad are\quad given,$$

then they can be expressed in the form of $$\frac { p }{ q } \quad As\quad 0.\overline { x } =\frac { x }{ 9 } \quad and\quad o.\overline { xy } =\frac { xy }{ 99 } $$

**(z)** The recurring decimal numbers of type $$0.\overline { x } \quad or\quad o.\overline { xyz } $$ may be converted to rational form as $$\frac { p }{ q } \quad follows.$$

$$0.\overline { xy } =\frac { xy-x }{ 90 } \quad or\quad o.\overline { xyz } =\frac { xyz-x }{ 990 } $$

Thanks you and All the best!

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