We'll discover what a real number is in this lesson. **The division algorithm of Euclid, the fundamental theorem of arithmetic,** **LCM** and **HCF** methods, and a thorough explanation of rational and irrational numbers with examples.

**The Basics of Real Numbers:**

All rational and irrational numbers are combined to form real numbers.

On the number line, any real number can be plotted.

For quick revision, students can use the links below to get the chapter's brief notes and MCQ questions, as well as a separate solution.

**The Division Lemma of Euclid**

Given two integers a and b, Euclid's Division Lemma says that there exists a unique pair of integers q and r such that a=bq+r and 0rb.

**Dividend = divisor quotient + remainder is roughly the same as this lemma.**

In other words, the quotient and remainder obtained for a given pair of dividend and divisor will be unique.

The Division Algorithm of Euclid

**The Euclid Division Algorithm is a method for determining the H.C.F of two numbers, such as a and b, where a> b.**

To find two integers q and r such that a= bq + r and 0rb, we use Euclid’s Division Lemma.

If r = 0, the H.C.F is b; otherwise, we use Euclid’s division Lemma on b (the divisor) and r (the remainder) to generate a new pair of quotients and remainders.

The process described above is repeated until the remaining is zero. In that phase, the divisor is the H.C.F of the supplied set of numbers.

**Arithmetic’s Fundamental Theorem**

**Prime Factorization** is a term that refers to the process of

The approach of representing a natural number as a product of prime numbers is known as prime factorization.

The prime factorization of 36, for example, is 36=2×2×2×3×3

**Arithmetic’s Fundamental Theorem**

The Fundamental Theorem of Arithmetic asserts that if the arrangement of the prime factors is ignored, the prime factorization for a given number is unique.

Visit here to learn more about the Fundamental Theorem of Arithmetic.

Least Common Multiple (L.C.M) Example: To find the Least Common Multiple (L.C.M) of 36 and 56, use the following method.

36=2×2×3×3

56=2×2×2×7

**HCF can be calculated using one of two methods: prime factorization or Euclid’s division technique.**

Prime Factorization is the process of expressing two numbers as products of their prime factors. Then we look for prime factors that are shared by both numbers.

As an example, The H.C.F. of 20 and 24 must be determined.

20 = 22.5 and 24 = 222.3

The factor that 20 and 24 have in common is 22. This equals 4, which in turn equals

The H.C.F of 20 and 24 is the result of this.

**The Division Algorithm of Euclid:**

It’s when you apply Euclid’s division lemma to find the H.C.F of two numbers over and over again.

Example: Class 10: Determine the HCF of 18 and 30 Real Numbers

**The HCF requirement is 6.**