Set of Natural Numbers.
N = {1, 2, 3, 4, ....}
* The smallest Natural number is '1' and the largest Natural number is not defined i.e., largest Natural number does not exists.
Set of Whole Numbers.
W = {0, 1, 2, 3, 4, ..........}
→ The smallest Whole Number is '0' and the Largest Whole Number is not defined.
→ The set of Whole Numbers contains all the Natural numbers.
→ The additional number in the set of Whole numbers compared to the set of Natural Numbers is '0'.
→ ∴ N ⊂ W
Set of Integers.
Z = {.... −3, −2, −1, 0, 1, 2, 3, ... }
The smallest integer is not defined and also the largest integer is not defined.
The set of Integers has all the Whole Numbers. In addition it has negative numbers.
∴ N ⊂ W ⊂ Z
Set of Rational Numbers
→ Neither the smallest nor the largest Rational Number exists.
→ As all the Integers can be written in
→ The set of Integers is the subset of Rational Numbers.
→ There can be many Rational Numbers between two Integers.
∴ N ⊂ W ⊂ Z ⊂ Q
Set of Irrational Numbers
The numbers that cannot be written in form where q ≠ 0 and p, q are Integers.
→ The set of Rational and Irrational numbers are different. They do not have any number in common.
→ can be written as that satisfies the criteria of writing in form q ≠ 0, but
p = is not an Integer. Hence is not a rational number. It is an Irrational.
→ The set of Irrational numbers is denoted by Q'
Set of Real Numbers
The Union of sets of Rational and Irrational numbers is the set of Real Numbers.
Terminating or Non - terminating Rational Numbers
When a rational number is written in its decimal form either it terminates or the decimal part recurres (i.e., repeats) if it is a non - terminating decimal.
e.g.: = 0.8 (terminates)
→ A number which is non - terminating and non - recurring is not a rational number i.e., we write a non - terminating non - recurring number in form, q ≠ 0 and p, q ∊ Z.
∴ A non - terminating, non - recurring decimal is an Irrational number.
The Fundamental Theorem, of Arithmetic
Every composite number can be written as a product of primes and this factorization is unique, apart from the order in which the prime factors occur.
Let us recall that
→ A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
→ A natural number greater than 1 that is not a prime number is called a composite number.
i.e., A composite number has atleast one positive divisor other than one and itself.
→ Consider the composite number 291060. Let us try to find its prime factors.
∴ 291060 = 2 × 2 × 3 × 3 × 3 × 5 × 7 × 7 × 11
= 22 × 33 × 5 × 72 × 11
As per the fundamental theorem of arithmetic 291060 can be written as product of primes and this is the unique way of writing the factorization of composite number as the product of prime numbers.
i.e., Whatever be the order of writing the prime numbers, we get the same primes with the same repetition.
Prime Factorization Method
L.C.M. (Least Common Multiple)
When two or more positive integers are written as the product of primes, their L.C.M. is the product of the greatest power of each prime factors in the numbers.
H.C.F. (Highest Common Factor)
When two or more positive Integers are written as the product of primes, their H.C.F. is the smallest power of each common prime factors in the numbers.
Consider three positive integers.
15, 40, 45
Let us write the prime factors of these numbers.
15 = 3 × 5
40 = 2 × 2 × 2 × 5 = 23 × 5
45 = 3 × 3 × 5 = 32 × 5
L.C.M. (15, 40, 45) = 23 × 32 × 5
= 8 × 9 × 5
= 360
[Product of the greatest powers of each prime factor i.e, 2, 3 and 5]
H.C.F. (15, 40, 45) = 5
[Product of the smallest power of each common prime factors in the number] 5 is the only common factor of all the numbers 15, 40 and 45.
Consider any two positive Integers 12, 15
12 = 2 × 2 × 3
= 22 × 3
15 = 3 × 5
L.C.M. (12, 15) = 22 × 3 × 5 = 60
H.C.F. (12, 15) = 3
Product of L.C.M. and H.C.F. = 60 × 3 = 180
Product of the numbers = 12 × 15 = 180
∴ We can conclude that
Product of two numbers = Product of their L.C.M. and H.C.F.
Note: This relation is limited to two positive numbers. It cannot be extended when we consider more than two numbers.
Application of Fundamental Theorem of Arithmetic.
→ When a number is written in exponential form.
e.g.: 6n where 'n' is a natural number. The concept of Fundamental Theorem of Arithmetic is applied to decide the last digit i.e, the digit with which the number ends for any natural number 'n'.
e.g.: Consider the number 6n where n is a natural number. Check whether there is any value of 'n' for which 6n ends with the digit 5.
Sol: For the number 6n to end with 5 for any natural number n, it should be divisible by 5
i.e, The Prime factorization of 6n must contain 5.
6n = (2 × 3)n
= 2n × 3n
2 and 3 are the prime numbers in the factorization of 6n
As 5 is not present in the prime factorization, there is no natural number 'n' for which 6n ends with the digit '5'.
Rational Numbers and their Decimal Expansions Theorem:
Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form , where p and q are coprime, and the prime factorization of q is of the from 2n5m where n, m are non - negative integers.
As per the theorem when the decimal expansion of a rational number terminate, it can be written in
e.g.: (1) We know that = 8.5 (a terminating decimal)
Let us try to write with the denominator 2 as 2n ⨉ 5m form.
e.g.: (2) can be written as 0.875 which is a terminating decimal
Right now, we have only 2 as the prime factor of the denominator 8.
Now Let's try to write 8 in the form of 2n ⨉ 5n i.e., Let's try to write an equivalent fraction to where 8 has both '2' and '5' as prime factor's
e.g.: (1) and (2) proves the Theorem.
[Formal proof is beyond the scope of the book. Hence examples are considered to prove it]
* The converse of the above theorem also holds good.
Theorem: Let x =
Theorem: Let x = be a rational number, such that the prime factorization of q is not of the form 2n ⨉ 5m where n and m are non - negative integers, then x has a decimal expansion which is non - terminating and repeating (i.e, recurring)
→ From the above two theorems we can conclude that, if a rational number's demoninator 'q' is written in 2n ⨉ 5m form it terminates otherwise the rational number is a non - terminating recurring decimal.
e.g.:
q = 7 is a prime number and cannot be written in 2n ⨉ 5m form
∴ Its decimal expansion do not terminate and repeats.
Let's check this fact by actual division
As again we have arrived at '1' the digits 142857 in the quotient repeats.
i.e., which is a non - terminating, recurring decimal.
* An Irrational number's decimal expansion is non - terminating, non - recurring.
e.g.: (1) 1.01001000100001....
(2) 2.345634567345678...
(3) 0.10110111011110...
The above three examples of decimal expansion are non - terminating, non - recurring.
∴ The above three examples represent Irrational number. Proof by contradiction is the technique used to prove a number to be Irrational
Prove that is Irrational
Proof: Let is Rational.
It can be written in form
= where q ≠ 0
If p and q have a common factor other than 1, then we divide by the common factor to get = where 'a' and 'b' are co-prime
b = a
Squaring on both the sides
3b2 = a2 ......... (1)
∴ 3 divides a2 as it is a factor of it
⇒ 3 divides a ............ (2)
[∴ If p is a prime and p divides a2 then it divides a]
a = 3c for some integer 'c'.
substituting a = 3c in (1)
3b2 = (3c)2
= 9c2
b2 = 3c2
⇒ 3 divides b2
⇒ 3 divides b ............ (3) [If p divides b2 then it divides b]
from (2) and (3) equations we find that the common factor of 'a' and 'b' which contradicts the fact that 'a' and 'b' are co-primes and have no other factor than 1.
This contradiction has arise because of the assumption that is rational.
∴ Our assumption is false and hence is irrational.
→ The sum or difference of a rational and Irrational number is Irrational.
e.g.: 2 + is Irrational.
Sol: Let 2 + is rational
Let a and b be two co-primes (b ≠ 0)
Such that 2 + =
⇒ 2 − =
∴ is rational
But this contradicts the fact that
∴ Our assumption 2 + is rational is false and hence, 2 + is Irrational
* The difference of a rational and Irrational number is Irrational.
E.g.: 3 − is Irrational.
Sol: Let 3 − be rational
Let 'a' and 'b' be two co-primes (b ≠ 0)
Such that 3 − =
⇒ 3 − =
⇒
This contradicts with the fact that is Irrational.
∴ Our assumption that 3 − is rational is false and hence 3 − is Irrational.
* The product of a non - zero rational and Irrational number is Irrational.
E.g.: 5 is Irrational.
Sol: Let 5 be Rational
Let 'a' and 'b' two co - primes (b ≠ 0)
Such that 5 =
=
As 5, a, b are integers,
But this contradicts with the fact that is Irrational
Our assumption 5 is rational is false and 5 is Irrational.
* The quotient of a non - zero rational and Irrational number is Irrational.
E.g.:
It can be proved as above using the proof by contradiction.
* The sum of two Irrational numbers need not be Irrational.
E.g.: (1) + is Irrational and
(2) + (− ) = 0 is Rational.
→ In the first example and are Irrational numbers and their sum
+ is Irrational.
→ In the second example and − are Irrational but their sum
→ As the sum of two Irrational numbers is not always Irrational we can conclude that
1. The sum of two Irrational numbers need not be Irrational.
2. The Product of two Irrational numbers need not be Irrational.
E.g.: (2): and are Irrational and their product × = is also an Irrational number.
→ As the product of two Irrational numbers is not always an Irrational number we can say that-
→ The Product of two Irrational numbers need not be Irrational.
But this contradicts the fact that is Irrational as the square root of a Prime number is Irrational.
∴ We can conclude that our assumption is false.
Hence is an Irrational number.
Properties of Real Numbers
You have studied the properties of Real Numbers in your earlier classes let us revise these properties.
For any three real numbers a, b, and c the following properties holds good.
Logarithms
Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. Tedious multiplication of multi - digit numbers can be replaced by simpler additions using logarithms.
The Present day notion of logarithms is given by Leonhar Euler who connected logarithms to the exponential function in the 18th century.
Let's Revise exponential function before defining logarithms.
Exponents
We know that 2 is the base and 8 is the Index.
Let us revise the Laws of Exponents
(i) am × an = am + n
(ii) am ÷ an = am−n, a ≠ 0, m > n
(iii) (am)n = amn
Definition of Logarithm
Consider an exponential equation
ay = x where x and a are positive real numbers and a ≠ 1
Then, the logarithmic form of ay = x is
logax = y
is read as "logarithm of x to base a is y" or in short ''Log of x to base a is y"
E.g.: 28 = 256 is the exponential form
log2256 = 8 is the logarithmic form
Laws of logarithms
→ Corresponding to each law of exponent, we can derive a law of logarithm writing it in logarithmic form.
(i) Suppose x = am and y = an where a > 0 and a ≠ 1
Writing in logarthmic form
logax = m and logay = n ......... (1)
We know that am × an = am+n
∴ xy = am × an = am + n
i.e, xy = am + n
Writing in logarithmic form
logaxy = m + n
= logax + logay [from (1)]
(ii) In the similar manner we can derive the second Law of logarithm
(iii) From the Law of Exponents (am)n = amn
we can derive the third Law of Logarithm
Expansion of logarithm using the Laws of Logarithm
→ Laws of Logarithm are used to write a Logarithm in expanded form.
While finding prime factors by factorization method
we observed 291060 = 22 × 33 × 5 × 72 × 11
= log (22 × 33 × 5 × 72 × 11) − log13
= log22 + log33 + log5 + log72 + log11 − log13
[∵ loga xy = loga x + loga y]
= 2 log2 + 3log3 + log5 + 2log7 + log11 - log13
Q. Write 5log2 + 3log7 − 2 log3 as a single logarithm
Sol: 5log2 + 3log7 − 2log3
= log25 + log73 − log32 [... logaxm = mloga x]
= log(25 × 73) − log32 [∵ logax + logay = logaxy]
Application of logarithm
As Logarithm is the means to simplify difficult calculations, it is widely used for all sorts of calculations in Engineering, business, economics and include calculating compound interest exponential growth and decay, pH value in chemistry, measurement of magnitude of earthquakes etc.
(Writer: V. Padmapriya)