Number system

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01 NUMBER SYSTEM

Theorem 1: Euclid’s Division Lemma:

G i v e n p o s i t i v e i n t e g e r s a a n d b, t h e r e e x i s t u n i q u e i n t e g e r s q a n d r s u c h t h a t, a = b q + r, 0 \leq r < b .

I n o t h e r w o r d s g i v e n i n t e g e r s a a n d b, w e c a n e x p r e s s a a s a = b \times Q u o t i e n t + r e m a i n d e r, 0 \leq r e m a i n d e r < b o r D i v i d e n d = D i v i s o r \times Q u o t i e n t + r e m a i n d e r

Euclid’s Division Algorithm:

It is application of Euclid’s Division Lemma

It gives us steps to find the HCF of two whole numbers as follows.

Given two whole numbers c and d, (c > d)

S t e p 1 : B y E u c l i d ’ s D i v i s i o n L e m m a, w e c a n w r i t e, C = d \times q + r, 0 \leq r < d, f o r s o m e q a n d r .

S t e p 2 : I f r = 0, d = (H C F (c, d) I f r \neq 0, a p p l y E u c l i d ’ s D i v i s i o n L e m m a t o t h e i n t e g e r s t o d a n d r .

S t e p 3 : C o n t i n u e t h i s p r o c e s s t i l l w e g e t r = 0 . T h e d i v i s o r a t t h i s s t a g e = H C F . F o r e g . H C F o f 12 a n d 16 . 16 = 12 \times 1 + 4 12 = 4 \times 3 + 0 R e m a i n d e r i s z e r o, ∴ H C F = 4

Theorem 2: The fundamental theorem of arithmetic:

E v e r y c o m p o s i t e n u m b e r c a n b e e x p r e s s e d a s a p r o d u c t o f p r i m e s, a n d t h i s p r i m e f a c t o r i s a t i o n i s u n i q u e a p a r t f r o m t h e o r d e r o f p r i m e s . e g . 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3

I r r a t i o n a l n u m b e r : N u m b e r s o f t h e t y p e \sqrt{p} (p i s n o t p e r f e c t s q u a r e) a r e c a l l e d i r r a t i o n a l n u m b e r s . e . g . \sqrt{2}, \sqrt{5}, π, 0.01010100 . . . .

Theorem 3:

L e t p i s p r i m e n u m b e r, I f p d i v i d e s a^{2}, t h e n p d i v i d e s a^{2} .

G i v e n : p d i v i d e s m^{2}, p = m a^{2}, m i s a n i n t e g e r .

T o p r o v e : p d i v i d e s a, i . e p = k a, f o r s o m e i n t e g e r k .

Proof:

L e t a = p_{1} p_{2} p_{3} p_{n} ., w h e r e p_{1_{}}, p_{2,} p_{3, . . . . . .} p_{n} a r e p r i m e s .

∴ a^{2} = {p_{1}}^{2} {p_{2}}^{2} {p_{3}}^{2} . . . . . . . . {p^{2}}_{n}

p d i v i d e s a^{2} ∴ p i s o n e o f p_{1}, p_{2}, p_{3}, . . . . . . p_{n}

S u p p o s e p = p_{1_{}}

∴ a = p_{1} p_{2} p_{3} . . . . . p_{n}

\Rightarrow p d i v i d e s a

Theorem 4:

\sqrt{2} i s i r r a t i o n a l n u m b e r .

G i v e n : a n u m b e r \sqrt{2} .

T o p r o v e : \sqrt{2} c a n n o t b e e x p r e s s e d a s \frac{p}{q} f o r m .

P r o o f : L e t ‘ s a s s u m e \sqrt{2} i s r a t i o n a l n u m b e r

\sqrt{2} = \frac{p}{q} (p a n d q d o n o t h a v e c o m m o n d i v i s o r)

∴ \sqrt{2} q = p

∴ 2 q^{2^{}} = p^{2} . . . . . . . . 1

\Rightarrow 2 d i v i d e s p^{2} .

\Rightarrow 2 d i v i d e s p . . . . . . (A)

∴ p = 2 d

\Rightarrow p^{2} = 4 d^{2} . . . . . . 2

\Rightarrow ∴ 2 q^{2} = 4 d^{2} . . . . . (f r o m 1 a n d 2)

\Rightarrow ∴ q^{2} = 2 d^{2}

\Rightarrow 2 d i v i d e s q^{2^{}}

\Rightarrow 2 d i v i d e s q . . . . . (B)

From and is common divisor of p and q.

T h i s c o n t r a d i c t s o u r a s s u m p t i o n .

S o o u r a s s u m p t i o n i s w r o n g .

∴ \sqrt{2} i s i r r a t i o n a l n u m b e r .

Theorem 5:

I f x h a s t e r m i n a t i n g d e c i m a l e x p a n s i o n .

I f X = \frac{p}{q}, t h e n q h a s p r i m e f a c t o r s o f t h e t y p e 2^{m} 5^{n}

Theorem 6:

I f x = \frac{p}{q}, a n d q h a s p r i m e f a c t o r s o f t h e t y p e 2^{m} 5^{n},

t h e n x h a s t e r m i n a t i n g d e c i m a l e x p a n s i o n .

Theorem 7:

: I f x = \frac{p}{q}, a n d q d o e s n o t h a v e p r i m e f a c t o r s o f t h e t y p e

2^{m} 5^{n}, t h e n x h a s n o n - t e r m i n a t i n g d e c i m a l e x p a n s i o n .

Next Chapter >

Polynomials

Pair of linear equation

Quadratic Equation

Arithmetic Progression

Triangles

Coordinate geometry

Introduction to trigonometry

Application of trigonometric ratio

Circles

Constructions

Areas related to circle

Surface areas and volumes

Statistics

Probability

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