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## Mathematical Statement (or Proposition)

A mathematical statement is a sentence which is either true or false . It can not be ambiguous.

## Conjecture

It is a conclusion or proposition based on incomplete information, for which no proof has been found.

## Axiom

A statement or proposition which is regarded as being established, accepted, or self-evidently true. It need not to be proved.

## Theorem

It is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms.

## Proof

The process which can establish the truth of a mathematical statement based purely on logical arguments is called a mathematical proof.

❕ The Fundamental Theorem of Arithmetic was first recorded as Proposition 14 of Book IX in * Euclid’s Elements*, before it came to be known as such.

It’s first correct proof was given by Carl Friedrich Gauss (1777-1855) in his *Disquisitiones **Arithmeticae* (1801).

**Carl Friedrich Gauss** is often referred to as the **‘*** Prince of Mathematicians’* and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science.

## The Fundamental Theorem of Arithmetic

*“Every composite number can be expressed ( factorised ) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.”*

➧ Given any composite number, there is **one and only one** way to write it as a product of primes, as long as we are not particular about the order in which the primes occur. e.g., we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written.

➧ In general, given a composite number *x*, we factorise it as , where are primes and written in ascending order, i.e. .

➢ When we combine the same primes, we get powers of primes. e.g.,

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 =

➢ Once we have decided that the order will be **ascending**, then the way the number is factorised, is **unique**.

## Applications of the Fundamental Theorem of Arithmetic

➧ The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields

➢ One of its application is in *Prime factorisation method* for finding HCF & LCM of two or more integers.

➢ It is applied *to prove the irrationality* of many irrational numbers.

➢ It is applied to explore when exactly the decimal expansion of rational number is *terminating* and when it is *non-terminating repeating**.*

### Finding HCF and LCM of Integers

➢ For any two positive integers p and q,

➢ For any three positive integers p, q and r,

(Q) Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

(A)

(common prime factors with their lowest power)

( all prime factors with their highest power)

### Proving Irrationality of an Irrational Number

To prove that a given number is irrational, we need the following theorem, whose proof is based on the Fundamental Theorem of Arithmetic.

#### Theorem : Let p be a prime number. If p divides a^{2}, then p divides a, where a is a positive integer.

**Proof : **

Let, , where are primes not necessarily distinct.

are prime factors of

*p* divides (given)

⇒ *p* is one of the prime factor of (from fundamental theorem of arithmetic)

But Prime factorisation of any number is unique

⇒ *p *is one of

∵

#### Theorem : √2 is irrational.

**Proof: **

Let us assume that √2 is rational, such that for integers *a *and *b *(*b≠0*), √*2 = a/**b *, where *a* and *b* are co-primes, i.e. ratio is in standard form.

Squaring both sides, we get

So, we can write for some integer *c*,

So, 2 is a common factor of *a* and *b*.

But this contradicts the fact that *a* & *b* are co-primes.

Hence, this contradicts our assumption that √2 is rational.

So, we conclude that √2 is irrational.

➤ The technique we used above is known as **Proof by Contradiction**.

### Irrationality of an expression with Rational and Irrational factors

➢ Sum and Difference of a rational number and a irrational number is irrational.

➣ The product and quotient of a non-zero rational number and irrational number is irrational.

Show that: 2−√3 is irrational.

Proof : Let us assume that 2-√3 is rational, such that for co-primes and , .

∵ and are integers

is rational

is rational

But this contradicts the fact that is irrational.

So, we conclude that is irrational.

## Rational Numbers & Their Decimal Expansions

Decimal expansion of a Rational number can be either

**→ ***terminating *or

**→ ***non-terminating reccurring (repeating).*

❔ Can we predict by looking at denominator when the given rational is terminating and when it is non-terminating reoccurring?

### Rational numbers whose decimal expansions are terminating

↬ Fraction whose denominator is power of 10 will terminate in its decimal expansion.

e.g., 375/100 = 3.75, 23/1000 = 0.023 etc.

∵ Prime factorisation of 10 = 2×5

⇒ Prime factorisation of ** , **where

**is any positive integer.**

*k*⇒ Denominator of a fraction which is a power of 10 can be expressed as .

If we simplify these fraction into a standard rational number, where *p *& *q* are co-primes then some powers of two’s or five’s or both in the numerator & the denominator will cancel each other out, then Prime factorisation of *q* will have the form , where *m* and *n* are some non-negative integers (can be 0).

e.g,

↬ Fraction whose denominator is power of 2 only will be terminating in its decimal expansion.

e.g., 27/2 = 13.5, 11/4 = 2.75, 7/8 = 0.875 etc.

↬ Fraction whose denominator is power of 5 only will be terminating in its decimal expansion.

e.g., 1/5 = 0.2, 200/125 = 1.6, etc.

#### Theorem:* *Let *x* = p/q be a rational number, such that the prime factorisation of q is of the form 2n×5m, where n, m are non-negative integers. Then *x* has a decimal expansion which terminates.

Example: Among following rational numbers, select which will terminate.

Solution:

(i) ,

Here,

will terminate and,

= 0.084375

(ii)

Here,

will terminate and,

= 0.9

(iii)

Here,

will not terminate.

(iv)

Here,

will not terminate.

#### Theorem : Let *x* be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q ,where p and q are co-prime, and the prime factorisation of q is of the form 2^{n}×5^{m}, where *n*, *m* are non-negative integers.

e.g, 375/1000 (= 0.375)

dividing by 125 (HCF), we get

We can convert the rational number in form of *p/q* to equivalent rational number of form *a/b* , where b is power of 10 .We can then easily write its decimal expansion which is terminating.

e.g,

(form of )

(multiplying both the numerator and denominator with )

### Rational numbers whose decimal expansions are non-terminating recurring (repeating)

Rational numbers in the form *p*/*q* where *q* is not of the form 2^{n}×5^{m}, will not have terminating decimal expansion.

e.g.,

#### Theorem: Let *x* = p/q be a rational number, such that the prime factorisation of q is not of the form , where n, m are non-negative integers. Then *x* has a decimal expansion which is non-terminating recurring (repeating).

e.g.,

Here,

will not terminate and,