Real Numbers (Part - 1)

Introduction Of Real Numbers

Real numbers are a fundamental concept in mathematics that forms the basis of much of our understanding of the world around us. In simple terms, real numbers are all the numbers that can be represented on the number line, including both rational and irrational numbers.

Rational numbers are numbers that can be expressed as the ratio of two integers, such as 1/2, 3/4, or -7/5. Irrational numbers, on the other hand, are numbers that cannot be expressed as the ratio of two integers, such as pi (3.14159265359...) or the square root of 2 (1.41421356...).

Real numbers are used extensively in many branches of mathematics, including algebra, calculus, geometry, and number theory. They are also used in many real-world applications, such as physics, engineering, and finance.

Real numbers are typically represented using decimal notation, which uses a decimal point to separate the integer and fractional parts of the number. For example, the real number 3.14159 represents pi to five decimal places. Real numbers can also be represented using scientific notation, which expresses the number as a power of 10 multiplied by a decimal number.

Euclid division algorithm

The Euclidean division algorithm is a method of dividing one integer (the dividend) by another integer (the divisor), and obtaining both the quotient and the remainder. The algorithm is based on the division algorithm, which states that given two integers a and b, with b not equal to zero, there exist unique integers q and r such that:

a = bq + r

where q is the quotient and r is the remainder.

The Euclidean division algorithm starts with the dividend and divisor, and repeatedly subtracts the divisor from the dividend until the remainder is less than the divisor. The last nonzero remainder is the greatest common divisor of the original dividend and divisor.

Here are the steps of the Euclidean division algorithm:

Let a be the dividend and b be the divisor.
Divide a by b to get the quotient q and remainder r.
If r is zero, then the algorithm stops, and the result is q.
If r is not zero, then set a = b and b = r, and go back to step 2.

The algorithm will terminate after a finite number of steps, because the remainder r decreases with each step, and is always positive. The greatest common divisor of the original dividend and divisor is the last nonzero remainder obtained in the algorithm.

Euclid's Division Lemma

Euclid's Division Lemma is a fundamental concept in number theory that states that given two positive integers "a" and "b", there exist unique integers "q" and "r" such that:

a = bq + r

where "q" is the quotient and "r" is the remainder obtained when "a" is divided by "b" and 0 <= r < b. In other words, "q" gives the number of times "b" divides "a" completely, and "r" gives the remaining part that is not divisible by "b".

For example, let's say a = 20 and b = 3. We want to find q and r such that 20 = 3q + r.

We start by dividing 20 by 3 to get:

20 = 3 × 6 + 2

Here, 6 is the quotient, and 2 is the remainder, which is less than 3. Therefore, we can write:

20 = 3 × 6 + 2

This means that 3 divides 20 six times completely, leaving a remainder of 2.

Euclid's Division Lemma is used in various mathematical proofs and applications, including the greatest common divisor (GCD), least common multiple (LCM), and modular arithmetic.

Milan Tomic

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