Real Numbers - Revision Notes

 CBSE Class–10 Mathematics

Revision Notes
CHAPTER 01
REAL NUMBERS

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• Natural numbers: Counting numbers are called Natural numbers. These numbers are denoted by N = {1, 2, 3, .........}
• Whole numbers: The collection of natural numbers along with 0 is the collection of Whole number and is denoted by W.
• Integers: The collection of natural numbers, their negatives along with the number zero are called Integers. This collection is denoted by Z.
• Rational number: The numbers, which are obtained by dividing two integers, are called Rational numbers. Division by zero is not defined.
• Coprime: If HCF of two numbers is 1, then the two numbers area called relatively prime or coprime.

1. Euclid’s division lemma :

For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying the relation a =bq +r, 0⩽r <b.$\text{a }=\text{bq }+\text{r},0⩽\text{r }<\text{b}.$.

Theorem: If a$a$ and b$b$ are non-zero integers, the least positive integer which is expressible as a linear combination of a$a$ and b$b$ is the HCF of a$a$ and b$b$, i.e., if d$d$ is the HCF of a$a$ and b$b$, then these exist integers x1$x_1$ and y1$y_1$, such that d=ax1+by1$d=a{x}_1+b{y}_1$ and d$d$ is the smallest positive integer which is expressible in this form.

The HCF of a$a$ and b$b$ is denoted by HCF(a,b)$\left(a,b\right)$.

2. Euclid’s division algorithms :

HCF of any two positive integers a and b. With a > b is obtained as follows:

Step 1 : Apply Euclid’s division lemma to a and b to find q and r such that

a =bq +r, 0⩽r <b.$\text{a }=\text{bq }+\text{r},0⩽\text{r }<\text{b}.$

b = Divisor

q = Quotient

r =  Remainder

Step II: If r  = 0, HCF (a,b)=b if r≠0$r\ne 0$, apply Euclid’s lemma to b and r.

Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

3. The Fundamental Theorem of Arithmetic :

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

Ex:24=2×2×2×3=3×2×2×2$Ex:24=2\times 2\times 2\times 3=3\times 2\times 2\times 2$

4. Let x=pq,$x=\frac{p}{q},$ q≠0$q\ne 0$  to be a rational number, such that the prime factorization of ‘q’ is of the form 2m+5n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating.

5. Let x=pq,q≠0$x=\frac{p}{q},q\ne 0$ be a rational number, such that the prime factorizationof q is not of the form 2m+5n, where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating.

6. p–√$\sqrt{p}$ is irrational, which p is a prime. A number is called irrational if it cannot be written in the form Pq$\frac{P}{q}$ where p and q are integers and q≠ 0.$\text{q}\ne 0.$

8. If a and b are two positive integers, then HCF(a, b) x LCM(a, b) = a x b

    i.e., (HCF x LCM) of two intergers = Product of intergers.

9. A rational number which when expressed in the lowest term has factors 2 or 5 in the denominator can be written as terminating decimal otherwise a non-terminating recurring decimal. In other words, if the rational number ab$\frac{a}{b}$ is, such that the prime factorization of b is of form 2m.5n,$2^m{.5}^n,$ where m and n are natural numbers, then ab$\frac{a}{b}$ has a terminating decimal expansion.

10. We conclude that every rational number can be represented in the form of terminating or non-terminating recurring decimal.