Basa Raja Gopal

Monday, August 24, 2020

Model problem-1: Prove that √2 is an irrational number-10th class mathematics-Proof of irrational number

Before going to learn the proof of irrational,

we know the following

Every rational number can be written as a fraction

let the simplest form of a rational number = p/q

Here p and q are co-primes and q≠0

Co-prime numbers

Two numbers are called “co-prime numbers” If their H.C.F. is “1”

Co-primes have NO common factor other than “1”

Co-primes have only one common factor that is “1”

Theorem

a is a positive integer and p is a prime number.

“If a² is divided by p then a is also divided by p”

Division algorithm

Dividend(a) = Divisor(b) x Quotient(q) + Remainder(r)

a=bq + r

If remainder(r) is “0”

a=bq

we said that "b divides a"

In this case:

“a” is the multiple of “b” and “q”

“b” and “q” are called factors of “a”

Contradiction Method

In this method, we take an assumption that “opposite to given statement”

After several steps, we get mathematical absurdity

Reason for this absurdity is our assumption

So our assumption is wrong

We conclude that given statement is correct.

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Prove that √2 is an irrational.

(We can prove this by contradiction method)

Proof:

Assume that √2 is not an irrational number

So √2 is a rational number

∴ √2 = p ⁄ q (Here p, q are co-primes and q≠0)

Squaring on both sides, we get

(√2 )² = ( p ⁄ q )²

2 = p²/q²

2q²= p²

p² = 2q²(LHS and RHS are interchanged)

2 divides p² So 2 divides “p”……..…......(i)

That means p is multiple of 2

let take p = 2k (k is an integer)

Substitute p = 2k in 2q²= p²

2q²= (2k)²

2q²= 4k²

q²= 2k²(divided by 2)

2 divides q², so 2 divides “q”………….(ii)

From (i) and (ii), we get

2 divides p and q

That means 2 is a factor of p and q

But p and q are co-primes

(They have only “1” as common factor)

Above two statements are contradict each other

Because of our assumption

So our assumption “√2 is not an irrational” is wrong

Therefore √2 is an irrational

Hence proved

Practice the following problems

1. Prove that √3 is an irrational

2. Prove that √5 is an irrational

3. Prove that √7 is an irrational

Click the link for "Proof of 5√2+7√3 is an irrational number"

Click the link for "Introduction of Sets"

Click the link for "Euclid division algorithm&HCF"

Monday, August 17, 2020

Introduction to Number System_Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers and Realnumbers

1. Natural numbers (N)

We use 1, 2, 3, 4, 5, ...... for counting of things, objects, persons... etc

These counting numbers are called Natural numbers.

These are denoted by N.

Natural numbers are start with 1.

So 1 is the smallest natural number.

There is No ending (Last) number in natural numbers.

Because,

When ever you count the number, one more number is exist.

So there is No greatest number in natural numbers.

∴ N = {1, 2, 3, 4, 5, 6, …………………. }

Every number is one more than its previous number except 1.

If we represent these numbers on a line, we get natural number line.

On this line, starting number is one.

For any two numbers, smaller number is lie left side of bigger number on this number line.

2. Whole Numbers (W)

Natural numbers including zero (0) are called Whole Numbers.

If we include zero to natural numbers we get whole numbers.

Whole numbers are denoted by W.

These numbers are start with 0.

So zero (0) is the smallest Whole number.

There is No ending (Last) number in Whole numbers also.

So there is No greatest number in Whole numbers.

∴ W = {0, 1, 2, 3, 4, 5, 6, …………………. }

Every number is one more than its previous number except 0.

Whole number set is contained all Natural numbers.

So Natural number set is Subset to Whole number set.

N ⊂ W

If we represent these numbers on a line, we get whole number line.

On this line, starting number is zero.

3.Integers (I/Z)

Combination of Positive numbers and Negative numbers including zero (0) are called Integers.

These are denoted by I or Z.

(Z is taken from the German word “Zahl”, which means number)

There is No starting (First) number and ending (Last) number in Integers.

So there is No smallest number and No greatest number in Integers.

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

Every negative number is less than Zero.

And every negative number is also less than positive numbers.

Integer number set is contained all Natural numbers and Whole numbers.

So Natural number set and Whole number set are Subsets to Integers.

N ⊂ W ⊂ Z

4.Rational numbers (Q)

The numbers, which can be written in the form of p/q (where p, q are integers and q≠0) are called Rational numbers.

These are denoted by Q.

Q = {p/q, p, q are integers and q≠0}

Every natural number can be written in the form of p/q (where q=1)

And Every whole number, Every Integer is also written in the form p/q (where q=1)

So, Rational numbers are contains all Natural numbers(N), all Whole numbers(W) and all Integers(Z)

N ⊂ W ⊂ Z ⊂ Q

All the fractions are belongs to Rational numbers(Q).

A fraction will be converted in to decimal form by "dividing numerator by denominator"

By this division we get decimal form of a fraction

These decimals are two types

i. Terminating decimal

ii. Non-terminating and recurring(repeating) decimal

And

Every terminating decimal and Non terminating-repeating decimal will be converted in to fraction form.

So Every terminating decimal and Non terminating-repeating decimal belongs to Rational numbers

5. Irrational numbers (Q^l)

The numbers, which cannot written in the form of p/q (where p, q are integers and q≠0) are called Irrational numbers.

These are denoted by Q^l.

All Non-terminating and Non-repeating decimals belongs to Irrational numbers(Q^l)

Real Numbers ®

Combination of Rational numbers (Q) and Irrational numbers (Q^I) are called Real Numbers. These are denoted by R.

Real Numbers for Class 10