Real Numbers

Positive integers, negative integers, irrational numbers, and fractions are all examples of real numbers. In other words, we can say that any number is a real number, except for complex numbers. Examples of real numbers include -1, ½, 1.75, √2, and so on.
In general,

• Real numbers constitute the union of all rational and irrational numbers.
• Any real number can be plotted on the number line.

Euclid’s Division Lemma

• Euclid’s Division Lemma states that given two integers a and b, there exists a unique pair of integers q and r such that $a=b\times q+rand0\le r<b$.
• This lemma is essentially equivalent to : dividend = divisor $\times$ quotient + remainder
• In other words, for a given pair of dividend and divisor, the quotient and remainder obtained are going to be unique.

Euclid’s Division Algorithm

• Euclid’s Division Algorithm is a method used to find the H.C.F of two numbers, say a and b where a> b.
• We apply Euclid’s Division Lemma to find two integers q and r such that $a=b\times q+rand0\le r<b$.
• If r = 0, the H.C.F is b; else, we apply Euclid’s division Lemma to b (the divisor) and r (the remainder) to get another pair of quotient and remainder.
• The above method is repeated until a remainder of zero is obtained. The divisor in that step is the H.C.F. of the given set of numbers.