1. Natural Numbers (N) :
Counting numbers are known as natural numbers. Thus 1, 2, 3, 4, …etc. are natural numbers.
˜ The first and the least natural number is 1 (one)
˜ Consecutive natural nos. differ by 1 (one).
2. Whole numbers (w) :
All natural numbers together with '0' form whole numbers. Thus 0, 1, 2, 3, 4, … etc. one are whole nos.
˜ The first and the least whole number is zero.
˜ Consecutive whole number differe by one.
3. Integers (I or Z) :
All natural nos. 0 and negative of natural nos. form integers for example. ……–4, –3, –2, –1, 0, 1, 2, 3, 4, … etc.
˜ O is neither a negative nor a positive number. It is a neutral no.
4. Prime numbers (P) :
A natural number, which is greater than 1 and divisible by one and by itself only, is called a prime number. For eg : 2, 3, 5, 7, 11, ……
˜ The smallest prime number is 2
˜ Except 2 ; all other prime nos. are odd.
5. Composite number (C) :
A natural number, which is greater than 1 and is not prime, is called a composite number. Thus 4, 6, 8, 9, 10, 12, 14, ……
˜ The smallest composite number is 4.
˜ A composite number can be even or odd.
˜ It has atleast three distinct factor.
6. Co-prime numbers :
If two numbers do not have any factor (other than 1) common; the numbers are said to be co-prime
Thus (i) 6 and 25 are coprime, no any common factor other than 1. (ii) 3 and 5 are co-prime, no any common factor other than 1.
˜ It is not necessary that any of the two co-prime numbers has to be prime also.
˜ All consecutive nos. are coprime.
7. Terminating decimals :
The decimal expansion ends after a finite number of steps of division. Such decimal expansions are called terminating decimals
For example : = 0.4, = 4.125 and so on.
8. Non-terminating decimals :
The decimal expansions never come to an end. Such decimal expansions are called non-terminating
For example = = 0.1818…, =0.3555……
9. Rational Numbers (Q) :
The numbers of the form , where p and q are integers and q ¹ 0, are known as rational numbers.
or
A number is rational if and only if its decimal representation is terminating or non-terminating but recurring
Ex. , 3, , 1.75, 1.666……, 4.23535, ……,
10. Irrational numbers :
A number which cannot be put in the form , where p and q are integers and q ¹ 0, is called an irrational number
or
A number whose decimal expression is non-terminating and non recurring is called an irrational number.
Eg : , , , + 2, , p, , ……
11. Non-terminating : Repeating (or Recurring) decimals :
A decimal in which a digit or a group of digits repeats continually or periodically is called a repeating or a recurring or a periodic decimal.
Ex : = 0.8333… = ; = 0.181818…… =
˜ Put a bar (_) above those digit/digits which are repeated.
12. Real Numbers (R) :
Rational numbers and irrational numbers taken together form real numbers.
13. Pure recrring decimal :
It is a decimal representation in which all the digits after the decimal point are repeated Eg: , , , ……
14. Mixed recurring decimal :
It is a decimal representation in which there are one or more digits present before the repeating digits.
Eg : , , , ……
15. Negative of an irrational number is an irrational number.
16. The sum or difference of a rational number and an irrational number is an irrational number.
17. The product of a non-zero rational number and an irrational number is an irrational number.
18. The sum, difference, product and quotient of two irrational numbers need not be an irrational number.
19. There are an infinite number of rational (irrational) numbers between two rational (or irrational) numbers.
20. If a is a rational number and n is a positive integer such that the nth root of a is an irrational number, then a1/n is called a surd eg. , , etc
21. If is a surd, or radical then 'n' is known as ordern or index of surd and 'a' is known as radicand.
22. A surd which has unity only as rational factor is called a pure surd. Eg. , , , , ……
23. A surd which has a rational factor other than unity is called a mixed surd. Eg . , , ……
24. Surds having same irrational factors are called similar or like surds.
25. Only similar surds can be added or subtracted by adding or subtracting their rational parts.
26. Surds of same order can be multiplied or divided.
27. If the surds to be multiplied or to be divided are not of the same order, we first reduce them to the same order and then multiply or divide.
28. The two irrational numbers whose product is a rational number, are called rationalising factor of each other. For eg : x – is called rationalising factor x +.
Similarly is a R.F. of Similarly is a R.F. of
29. The surds which differ only in sign (+ or –) between the terms connecting them, are called conjugate surds eg. and or 2 + and 2 – are conjugate surds (binomial).
˜ Sum and product of two cojugate binomial factors are always rational numbers.
30. Laws of exponents for Real numbers :
(i) am × an = am + n (ii) (am)n = amn (iii) = am – n ; m > n
(iv) am × bm = (a × b)m (v) a–m = or = am,
(vi) (a × b)m = am × bm (vii) if a ¹ 0 (viii) aº = 1
(ix) (1)p = 1 where p is any rational no (x) If a ¹ 1 and ap = aq then p = q where p & q are rational nos
(xi) where a is any rational no. (xii) (–a)m = am, if m is even and (–a)m = –am, if m is odd.
Ex.1 Is zero a rational number ? Can you write it in the form , where p and q are integers and q ¹ 0 ?
Sol. Yes, zero is a rational number. We can write zero in the form whose p and q are integers and q ¹ 0.
so, 0 can be written as etc
Ex.2 Find six rational numbers between 3 and 4.
Sol. Hint : first rational number between 3 and 4
=
Ex.3 Find five rational numbers between and .
Sol. Hint : Let a =
d = =
so, Rational number are
a + d, a + 2d, a + 3d........
Ex.4 State whether the following statements are true or false ? Give reasons for you answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Sol. (i) True, the collection of whole number contain all natural number.
(ii) False, –2 is not whole number
(iii) False, is a rational number but not whole number.
Ex.5 State whether the following statements are true or false ? Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form , where m is a natural number.
(iii) Every real number is an irrational number.
Sol (i) True, since collection of real number consist of rational and irrational.
(ii) False, because no negative number can be the square root of any natural number.
(iii) False, 2 is real but not irrational.
Ex.6 Are the square roots of all positive integers irrational ? If not, give an example of the square root of a number that is a rational number.
Sol. No, is a rational number.
Ex.7 Write the following in decimal form and say what kind of decimal expansion each has :
(i) (ii) (iii) (iv) (v) (vi)
Sol. (i)
(ii)
(iii) = 4.125 (terminating decimal)
(iv) = 0.230769230769......
= (Non Terminating repeating)
(v)
(Non Terminating repeating)
(vi) = 0.8225 terminating
Ex.8 Classify the following numbers as rational or irrational :
(i) (ii) (iii) (iv) (v) 2p
Sol. (i) Q 2 is a rational number and is an irrational number
\ 2 – is an irrational number.
(ii)
Þ = 3 is a rational number.
(Rest Try Yourself)
Ex.10 Recall, p is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, . This seems to contradict the fact that p is irrational. How will you resolve this
contradiction ?
Sol.. which is approximate value of p
Ex.12 Find :
(i) (64)1/2 (ii)321/5
Sol.. (i) (64)1/2 = = = 81 = 8
(ii) 321/5 = (25)1/5 = = 21 = 2
(Rest Try Yourself)
Ex.13 Find :
(i) 93/2 (ii)322/5
Sol.. (i) = (3)3 = 27
(ii) = 22 = 4
(Rest Try Yourself)
Ex.15 Insert 4 rational numbers between and .
Sol. As numbers to be inserted are more than 3, we would follow method II., (Method I, a < < b)
Here the numbers given are and both of which have the same denominator.
\ We multiply numerator and denominator of each number by (4 + 1) = 5
to get and or and . Any 5 integers between 10 and 25 are 11, 12, 13, 14, 15.
\ Required rational numbers between the two given numbers are .
Ex.17 Convert into the form .
Sol. The given number is = 0.7283283 ….
Let, x = 0.7283283 … …(1)
Here after decimal there is only one digit namely 7, which is not recurring.
We multiply both sides of equation (1) by 10 to get 10 x = 7.283283… …(2)
Now after decimal 3 digits are recurring (283).
We multiply both sides of equation (2) by 1000 to get, 10000 x = 7283.283… …(3)
Subtracting equation (2) from equation (3), we get 90 x = 7276
Þ x = = which is the required form of the number.
Ex.18 Write 3 irrational number between 4.75 and 4.76.
Sol. Keeping in mind that decimal representation of an irrational number is neither terminating nor recurring, we can write any three numbers between 4.75 and 4.76 whose decimal representation is neither terminating nor recurring e.g., 4.7513428965832…, 4.7523471098623…, 4.7534829153785… .
Ex.19 Locate on number line.
Sol. We know that 5 = 22 + 12. So on real number line X'OX, take a point A so that OA = 2 units. At A, draw a ray AY1 perpendicular to real number line. Now with A as centre and 1 unit as radius draw an arc intersecting ray AY1 at B1. Join OB1. With O as centre and OB1 as radius draw an arc intersecting number line at P1. P1 is the point on number line representing i.e., OP1 = .
Fig. 11 Representing on number line.
Now at P1 draw ray P1Y2 perpendicular to number line and with P1 as centre and 1 unit as radius draw an arc intersecting P1Y2 at B2. Join OB2. With O as centre and OB2 as radius draw an arc intersecting the number line at P2. P2 is the point representing the location of . Again at P2 draw a ray P2Y3 perpendicular to number line and cut an arc at B3 on it with arc radius 1 unit and centre as P2. Join OB3. With O as centre and OB3 as radius draw another arc intersecting the number line at P3. P3 is the point corresponding to .
Ex.20 With the help of examples show that the quotient of two irrational numbers can be rational or irrational.
Sol. Consider two irrational numbers a = and b = then their quotient which is rational, while if we take two numbers as c = and d = both of which are irrational then their quotient which is an irrational number.
Ex.21 Locate 4.683 on number line by the method of successive magnification.
Sol. Lie between 4–5, 4.6–4.7, 4.68–4.69.
Visualization of 4.683 on number line.
Ex.22 If = 23x – 10. Find the value of x, given that x ¹ 10.
Sol. Þ = 23x – 10 Þ = 23x – 10 Þ = 23x – 10
Þ = 23x – 10 Þ = 23x – 10 Þ = 23x – 10
Þ 2x – 2 = 23x – 10 Þ x – 2 = 3x – 10 Þ 2x = 8
Þ x = 4.
Ex.23 If 2x = 5y = 10z, then prove that .
Sol. Let 2x = 5y = 10z = K. Þ 2 = K1/x, 5 = K1/y, 10 = K1/z
Now we know that 2 × 5 = 10 Þ Þ Þ .
Ex.24 If x = + 1, find the value of
Sol. x = + 1 Þ
\ = 4 × 3 = 12.
Ex.25 If x = 2 + , find the value of x2 + .
Sol. x = 2 + Þ \
Also \ = 42 – 2 = 16 – 2 = 14.
Ex.26 If x = and y = , find the value of 3x2 + 4xy + 3y2.
Sol.
\ x + y = Also, xy = = 1
Hence 3x2 + 4xy + 3y2 = 3(x2 + y2) + 4xy
Ex.27 If x = , find the value of x2 + 4x – 1 and x3 – 2x2 – 25x + 7.
Sol. x = Þ x + 2 = Þ (x + 2)2 =()2 Þ x2 + 4x + 4 = 5
Þ x2 + 4x – 1 = 0 Also x3 – 2x2 – 25x + 7 = (x2 + 4x – 1) (x – 6) + 1
(Here we observe that if (x3 – 2x2 – 25x + 7) is divided by x2 + 4x – 1, quotient is x – 6 and remainder = 1. So we can use dividend = divisor × quotient + remainder, to get the above relationship.)
\ x3 – 2x2 – 25x + 7 = 0 × (x – 6) + 1 = 1.