1. Subject :
Algebra
Std - 9th
Subject- Algebra
Topic- Real Numbers

2. Topic :
Real Numbers
Std - 9th
Subject- Algebra
Topic- Real Numbers

3. Sub-Topics : Std - 9th Subject- Algebra Topic- Real Numbers Revision.
Definition of real numbers.
Properties of real numbers.
Representation of rational number on the number line.
Representation of irrational numbers on the number line.
Euclid’s division lemma.
The fundamental theorem of Arithmetic.
Surds.
Forms of surds.
Similar surds or like surds.
Operation on surds.
Rationalization of surds.
Binomial expression of a quadratic surd.
Std - 9th
Subject- Algebra
Topic- Real Numbers

4. REAL NUMBERS
*Classify the following numbers into five groups natural numbers, whole numbers, rational numbers and
Irrational numbers .
2/3, 7, -4/5, 0, -10, 0.3, √2, √3, …… .
Std - 9th
Subject- Algebra
Topic- Real Numbers

5. Std - 9th Subject- Algebra Topic- Real Numbers
Classification
Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers
Ameya 7
-10
0.3 √2
Bismilla
0,7
0,7,-10
2/3,-0.3
√2, √3
Chaitanya
0,7,-10,2/3,-4/5,0.6
0.1011…
Let us compare these answers :
Here we observe that thinking process of Chaitanya about number systems is more clear than others.
Std - 9th
Subject- Algebra
Topic- Real Numbers

6. Natural numbers:-The counting numbers 1,2,3,4, … are called natural numbers. We write them as a set N = {1,2,3,4, …}
Is the sum of two natural numbers a natural number ?
Dolly says, "yes.”
Since 4 is N, 7 is N, = 11 is N .
Is the product of two natural numbers a natural number ?
Elizabeth says, ”yes.”
Since 5 is N and 8 is N, 5 x 8 = 40 is N
Std - 9th
Subject- Algebra
Topic- Real Numbers

7. Is the subtraction of two natural numbers a natural number ?
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Is the subtraction of two natural numbers a natural number ?
Fakir says, “No.”
Since 3 is N , 7 is N, 3-7 = -4 N
What do we conclude from above three examples?
Teacher says that the set of natural numbers is adequate for addition and multiplication but not for subtraction.
Std - 9th
Subject- Algebra
Topic- Real Numbers

8. Whole Numbers : The union of set of natural numbers and zero is a set of WHOLE NUMBERS & the set is denoted by ‘W’.
W = {0,1,2,3,4, }
Integer : The set of natural numbers, zero and opposite of all natural numbers is called as set of integers and denoted by I.
I = { ,-1,0,1, 2, }
Std - 9th
Subject- Algebra
Topic- Real Numbers

9. Is the division of two integers a integer? Govind says, “No.”
Since 2 is I, 5 is I, but 2/5 is I.
Rational Number : If p & q are integers (q‡0) then the number p/q is called a rational number and the set of rational number is denoted by Q.
Q = { /2,0,1/ }
Std - 9th
Subject- Algebra
Topic- Real Numbers

10. Std - 9th Subject- Algebra Topic- Real Numbers
Are all the natural numbers a rational number ?
Hemant says, “yes.”
Since 2 is N and 2= 2/1 is I
Are all the whole numbers a rational number ?
Indu says, “yes.”
Since 0 is W and 0= 0/5 is I
Are all the integers a rational number ?
Jaheer says, ”yes.”
Since –3 is I and -3= -3/1 is I
Rational numbers include natural numbers, whole numbers and integers
Std - 9th
Subject- Algebra
Topic- Real Numbers

11. *Natural numbers:- The smallest natural number is 1, and the largest number cannot be defined.
*Whole numbers:- The smallest whole number is 0.
*Integers:-The German word “Zahlen” means to count and “zahl” means number.
*Rational number:- The word rational is derived from the word “Ratio.”
Std - 9th
Subject- Algebra
Topic- Real Numbers

12. *Equivalent numbers:- 1/3 = 2/6 = 3/9…… These are equivalent numbers.
*1/2=0.5, 1/3=0.333…, 1/4=0.25, 1/5=0.2,
1/6= …, 1/10=0.1.
NOTE:-Rational whose denominators factors are 2 ,5 or 2 and 5 are terminating decimals.
*Terminating decimals and non terminating recurring decimals are both rational numbers.
*Rational numbers include natural numbers, Whole numbers and integers.
Note : Terminating decimals and non-terminating recurring decimals are both rational numbers.
Std - 9th
Subject- Algebra
Topic- Real Numbers

13. TO FIND RATIONAL NUMBERS BETWEEN TWO RATIONALS
If a & b are two rational numbers then (a+b)/2 is between a and b.
For ex. Between 3 and 4 there is a number (3+4)/2, i.e., 7/2.
Between 3 and 7/2 there is a number (3+7/2)/2=13/4.
Also, between 3/2 and 4 there is a number (3/2+4)/2=11/4.
. CONCLUSION : There are infinite rational numbers between two rational numbers
Std - 9th
Subject- Algebra
Topic- Real Numbers

14. *Archimedes was a Greek genius who first calculated the digits in decimal expansion of π. He found the value of π lies between and
< π <
*Aryabhatta was the greatest Indian mathematician who found the value of π correct to four decimal places.
π = (approximately)
Std - 9th
Subject- Algebra
Topic- Real Numbers

15. Conversion of terminating decimals into form m/n.
(i) Remove the decimal point from the numerator and write 1 in the denominator and put as many zeros on the right of 1 as the numbers of digits after the decimal point.
(ii) Convert the rational number obtained into its lowest term by dividing the numerator and the denominator by the common factor.
For ex. Convert 0.45 into form m/n.
0.45 = 45/100 = 9x5/20x5 = 9/20.
Std - 9th
Subject- Algebra
Topic- Real Numbers

16. CONVERSION OF NON-TERMINATING RECURRING DECIMAL NUMBER
Put the given decimal number equal to x.
Remove the bar if any and write the recurring digits at least twice.
If the recurring decimal has one place repetition multiply by 10, if there is two place repetition multiply by 100 and so on.
Subtract the number in (ii) from the number obtained in step (iii).
Divide both sides of equation by the coefficient of x.
Std - 9th
Subject- Algebra
Topic- Real Numbers

17. RULE FOR CONVERSION OF A PURE RECURRING DECIMAL IN THE FORM m/n.
Write the given number after removing decimal as the numerator.
In the denominator write down as many 9’s as the number of repeating digits.
For ex.
Express 0.7 in the form m/n.
Solution:- Write 7 as the numerator and one nine in the denominator (as only 1 repeating digit)
0.7 = 7/9.
Represent in the form m/n /99
Std - 9th
Subject- Algebra
Topic- Real Numbers

18. IRRATIONAL NUMBERS
The numbers which cannot be expressed in decimal form either in terminating or non-terminating recurring decimals are known as irrational numbers.
An irrational number cannot be written in the form m/n, where m and n are both integers and n ¹ 0.
Std - 9th
Subject- Algebra
Topic- Real Numbers

19. PYTHAGORAS
In 400 B.C . Pythagoras was Greek mathematician who was the first to discover the numbers which were not rational. These numbers are called irrational numbers.
REAL NUMBERS: Rational numbers and irrational numbers taken together are known as real numbers.
Std - 9th
Subject- Algebra
Topic- Real Numbers

20. REPRESENTATION OF IRRATIONAL NUMBERS ON THE NUMBER LINE.
Std - 9th
Subject- Algebra
Topic- Real Numbers

21. DIVISION ALGORITHM
We know that division is not defined in set of integers . For ex. If we divide 22/3, we get 7 as the quotient and 1 as the remainder. Thus, 22=3 x This is division algorithm.
Std - 9th
Subject- Algebra
Topic- Real Numbers

22. Symbolically we state division algorithm as : a and b are integers and b > 0. There exists a unique pair of integers q and r, such that a=bq +r where 0< r < b. In arithmetic the students have seen the truth of this theorem. Here we see that a is the dividend, b is the divisor, q is the quotient and r is the remainder.
Std - 9th
Subject- Algebra
Topic- Real Numbers

23. EXAMPLE 12=q 283 23 -23 53 -46 7 = r Std - 9th Subject- Algebra
Now let a = 283, b = 23. Clearly, the remainder 7 lies between 0 and 23, i.e., 0 < 7 < 23 implies 0 < r < 23. Such types of 0examples help us to have DIVISION ALGORITHM In the form a = bq + r for all 0 < r < b. This theorem is very useful to find the HCF of two numbers a and b which is written as HCF of (a , b) = c or simply we write (a, b)=c Note(a,b)=c means HCF of two numbers a and b is c.
12=q 23
283
-23 53
-46
7 = r
Std - 9th
Subject- Algebra
Topic- Real Numbers

24. Example to find the HCF of 240, 128.
Solution:- 240=128 x Here the divisor is 128 and remainder is 112. Apply the division algorithm again on 128 and =112 x Now the divisor is 112 and the remainder is 16. We again apply the division algorithm on 112 and = 16 x Now remainder is 0 and HCF of 112 and 16 is 16. Therefore we can say that HCF of 240 and 128 is 16. Here HCF(240,128)= HCF of (128,112)= HCF of (112,16)=16.
Std - 9th
Subject- Algebra
Topic- Real Numbers

25. FUNDAMENTAL THEOREM OF ARITHMETIC
We know that any natural number can be expressed as a product of primes and this is unique apart from the order in which the prime factors occur. For ex. , 3 = 3 x 1, 4 = 2 x 2 = 22 , 6 = 2 x 3, 8 = 2 x 2 x 2 = 23, 36 = 2 x 2 x 3 x 3 = 22 x 32, 192 = 8 x 24 = 4 x 2 x 4 x 6 = 2 x 2 x 2 x 2 x 2 x 2 x 3 = 26 x 3 In general when a composite number decomposes into primes and are written in ascending order.
Std - 9th
Subject- Algebra
Topic- Real Numbers

26. FUNDAMENTALTHEOREM
Every composite number can be expressed as a product of prime numbers and this factorization is unique apart from the order in which the prime factor occur .
This fundamental theorem of arithmetic helps us to find H.C.F and L.C.M of numbers.

27. WORKING RULE TO FIND H.C.F
Select the lowest of the powers of common primes.
The product of powers of common prime is H.C.F.
Factorize the numbers into primes.
WORKING RULE TO FIND L.C.M
1] Factorize the numbers into primes.
2] Select the highest power of a prime present in all or some of the numbers.
3] The product of powers of common and uncommon prime is L.C.M.

28. Determine H.C.F. and L.C.M. of 120, 90
Solution:- 120 = 8 x 15 = 2 x 2 x 2 x 3 x 5 = 23 x 3 x 5 90 = 6 x 15 = 2 x 3 x 3 x 5 = 2 x 32 x 5 H.C.F. = 21 x 31 x 51 = 30

29. DETERMIN H.C.F AND L.C.M OF 8, 25
Solution:- 8 = 2 x 4 = 2 x 2 x 2 = = 5 x 5 = 52 H.C.F of (8, 25) = 1 **No common factor between 8 and 25 L.C.M of (8, 25) = 8 x 25 = 200
NOTE:
H.C.F of any two or more prime numbers is always 1.
L.C.M of any two or more prime numbers is equal to their products.

30. SURDS
If n is a natural number (n > 1) a is a positive rational number, and n√a is an irrational number is called a surd. n is called order of surd, a is called the radicand and the symbol √ is a radical sign.

31. LAWS OF SURDS
If a and b are rational a, b > 0 and m, n, p are natural numbers then:
(n√a)n = a
(n√a)(n√b) = (n√ab)
(n√a)/(n√b)=(n√(a/b))
(n√(m√a))= (m√(n √a)) = mn√a
(m√an) = (mp√anp)
(m√an) = (m√a)n

32. FORMS OF SURDS
Mixed Surds- A surd of the form k n√a where k is a rational number, k ¹ 0 and k ¹ ±1 is called a mixed surd.
For ex. 2 3√7, ¾ √5 etc are mixed surds.
Pure Surds- A surd of the form k n√a where k is a rational number, k = ±1 is called a Pure surd.
For ex. √3, √5, 3√2 etc are Pure surds.
NOTE: A mixed surd can be expressed as a pure surd.
For ex. 5 √2 = √25 x √2 = √50

33. SIMILAR SURDS
The surds of the form pn√a and qn√a where p and q are rational numbers are called similar surds or like surds. For ex. 5√2, 7√2, 3√2/4, are similar surds. √12, √50 are also similar surds.

34. COMPARISON OF SURDS For ex. If n√a and n√b are two surds then:
Two surds of the same order can be compared by comparing their radicands.
For ex.
If n√a and n√b are two surds then:
n√a = n√b, if a = b.
n√a > n√a, if a > b.
n√a < n√b, if a < b.

35. METHOD TO COMPARE SURDS
Convert each surd into the same order.
Compare the radicands obtained.
A surd with larger radicand is larger of the given surds.
For ex. √3, 3√2
√3 = 6√33 = 6√27 and 3√2 = 6√22 = 6√4
But 6√27 > 6√4.
Therefore √3 > 3√2

36. OPERATION ON SURDS ADDITION AND SUBTRACTION OF SURD
We know that the operations of addition and subtraction are defined on real numbers. As surds are irrational numbers they can be added or subtracted but addition and subtraction can be done only on like surds.
Is sum of two surds a surd ?
For ex. 5√2 + 3√2 = 8√2 (5+√2) +(2- √2)=7
7√3 - 2√3 = 5√3

37. MULTIPLICATION AND DIVISION OF SURDS: Using laws of surds, surds can be multiplied or divided. Note that multiplication and division of surds be performed on the surds of same order . Is product /division of two surds a surd ? For ex. √7 x √3 = √21 , 2 √3 x √3 =6 , 5 √3 / 2 √3 = 5/2 , 5 √6/ √2=5 √3 …

38. RATIONALIZATION OF SURDS
If the product of two surds is a rational number then each surd is called a rationalizing factor of the other surd.
For ex. √24 x √6 = √144 = 12
√24 x √24 = 24
Therefore √6 is a rationalizing factor of √24. Also, √24 is a rationalizing factor of √24.
NOTE: The rationalizing factor of a given surd is not unique but it is convenient to use the lowest form of rationalizing factor of a surd
Std - 9th
Subject- Algebra
Topic- Real Numbers

39. The binomial expressions of quadratic surds
a + y√b and a-y √b are said to be conjugate of each other.
For ex. 2 √2- √3 and 2 √2 + √3 are conjugate to each other.
Note: Conjugate surds are useful for rationalization since the product of two conjugate surds is a rational number.
For ex. (√3 + √2)(√3- √2) = (√3)2 – (√2)2 = = 1
Std - 9th
Subject- Algebra

40. BINOMIAL EXPRESSION OF A QUADRATIC SURD
The sum of two numbers one of which is a quadratic surd and the other, either a non-zero rational number or a quadratic surd is called Binomial expression of quadratic surd.
The general binomial expression of quadratic surd is a+ y √b where a is a non-zero rational number or a quadratic surd, y is a non-zero rational number and √b is a quadratic surd.
For ex. √5 - √2, √7 + √11, 3 + √5, 5 - √7 are binomial expressions of quadratic surds.

41. Exercise Topic- Real Numbers
Write down the rationalizing factor of 3√625.(2 mark)
Find the rational number between 3 and 4. (1 mark)
Express … in the form m/n (1 mark)
Express … in the form m/n (2 marks)
Explain which is greater √17 - √12 or √11- √8.(3 mark).
Solve 5x = (2 marks)
Find square root of 10+√24 + √60 + √ (4 mark)
Represent 1/ √7 on a number line (5 mark)
Topic- Real Numbers

42. Exercise Que.1) Express each of the following in the form of p . q
0.555…
0.4747…
Que.2) Write the following real numbers in decimal form.
Que.3) Represent √10 on the number line.
Que.4) Show that 7+ √7‾ is an irrational number.
Que.5)Use Euclid’s division lemma to find the HCF of the following.
1.) 16 and ) 40 and 248

43. Creator :
Fulsundar G. T.
B.Sc.B.Ed
S.T.V.Dehu