## Pre-Requisires

Test & Enrich

**Real Numbers | Speed Notes**

**Notes For Quick Recap**

**Introduction:**

**Euclid’s Division Lemma/Euclid’s Division Algorithm :**

Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.

This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder. **(Scroll down to continue …)**.

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**Introduction:**

**Euclid’s Division Lemma/Euclid’s Division Algorithm:**

Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.

This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder.

**NOTE:**

1. Lemma is a proven statement used for proving another statement.

2. Euclid’s Division Algorithm can be extended for all integers, except zero i.e., b 0.

**HCF of two positive integers :**

HCF of two positive integers a and b is the largest integer (say d ) that divides both a and b(a>b) and is obtained by the following method :

**Step 1.** Obtain two integers r and q, such that a=bq+r, 0r<b.

**Step 2.** If r=0, then b is the required HCF.

**Step 3.** If r0, then again obtain two integers using Euclid’s Division Lemma and continue till the remainder becomes zero. The divisor when remainder becomes zero, is the required HCF.

**The Fundamental Theorem of Arithmetic :**

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

**Irrational Number :**

A number is an irrational if and only if, its decimal representation is non-terminating and non-repeating (non-recurring).

OR

A number which cannot be expressed in the form of pq , q 0 and p, qI, will be an irrational number. The set of irrational numbers is generally denoted by Q.

**NOTE:**

1. The rational number pq will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2^{n}5^{m}, where n, m are some non-negative integers.

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