Number
A number is a mathematical object used to count, measure, and label.
Real number=R (Generally denoted by R):
All rational and irrational numbers are called real numbers.
Any real number can be plotted on the number line.
Q = Rational Numbers:
It is in the form p/q where, q ≠ 0, p, q ∈ I are rational numbers. (“∈” means “belongs to”)
All integers can be expressed as rational numbers, for example, 5 = 5/1.
And having a decimal expansion of rational numbers terminating or non-terminating recurring.
Q’ = Irrational Numbers:
Real numbers which cannot be expressed in the form p/q and whose decimal expansions are non-terminating and non-recurring.
Roots of primes like √2, √3, √5, etc. are irrational.
N = Natural Numbers:
Counting numbers are called natural numbers. N = {1, 2, 3, …}.
W = Whole Numbers:
Zero along with all-natural numbers are together called whole numbers {0, 1, 2, 3,…}.
Even Numbers:
Natural numbers of the form 2n are called even numbers. (2, 4, 6, …}
Odd Numbers:
Natural numbers of the form 2n -1 are called odd numbers. {1, 3, 5, …}
Important Points:
All Natural Numbers are whole numbers.
All Whole Numbers are Integers.
All Integers are Rational Numbers.
All Rational Numbers are Real Numbers.
Prime Numbers:
The natural numbers greater than 1 are divisible by 1 and the number itself only is called a prime number.
Prime numbers have two factors i.e., 1 and the number itself, For example, 2, 3, 5, 7 & 11, etc.
Composite Numbers:
The natural numbers which are divisible by 1, itself and any other number or numbers are called composite numbers. For example, 4, 6, 8, 9, 10, etc.
Note: 1 is neither prime nor a composite number.
Also, 1 is not a prime number as it has only one factor.
Euclid’s Division Lemma:
For a given positive integers a and b,
there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
As we know “dividend = divisor × quotient + remainder”.
It also means that for a given pair of dividends and divisors, the quotient and remainder obtained are going to be unique.
Euclid’s Division Algorithm:
To obtain the HCF of two positive integers, say c and d, with c > d, follow
the steps below:
Step 1: Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and
r such that c = dq + r, 0 ≤ r < d.
Step 2: If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.
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.
Method of Finding HCF:
H.C.F can be found using two methods –
(1) Prime factorisation and
(2) Euclid’s division algorithm.
Prime Factorisation:
Given two numbers, we express both of them as products of their respective prime factors. Then, we select the prime factors that are common to both the numbers
Example – To find the H.C.F of 20 and 24
20=2×2×5 and 24=2×2×2×3
The factor common to 20 and 24 is 2×2, which is 4, which in turn is the H.C.F of 20 and 24.
Euclid’s Division Algorithm:
It is the repeated use of Euclid’s division lemma to find the H.C.F of two numbers.
Important:
“For any two positive integers a and b,
a×b=H.C.F×L.C.M.”
Remarks:
- Euclid’s division lemma and algorithm are so closely interlinked that people often
call former as the division algorithm also. - Although Euclid’s Division Algorithm is stated for only positive integers, it can be
extended for all integers except zero, i.e., b ≠ 0. However, we shall not discuss this
aspect here.
Example: Find the LCM and HCF of 6 and 20 by the prime factorization method.
Solution : We have : 6 = 2^1× 3^1 and 20 = 2 × 2 × 5 = 2^2×5^1
Note that
HCF(6, 20) = 2^1= Product of the smallest power of each common prime factor in the numbers.
LCM (6, 20) = 2^2× 3^1× 5^1= Product of the greatest power of each prime factor, involved in the numbers.
From the example above, you might have noticed that HCF(6, 20) × LCM(6, 20)= 6 × 20. In fact, we can verify that for any two positive integers a and b,
HCF (a, b) × LCM (a, b) = a × b. We can use this result to find the LCM of two positive integers if we have already found the HCF of the two positive integers.
Rational and Irrational Numbers:
If a number can be expressed in the form p/q where p and q are integers and q ≠ 0, then it is called a rational number.
If a number cannot be expressed in the form p/q where p and q are integers and q ≠ 0, then it is called an irrational number:
If p (a prime number) divides a^2 , then p divides a as well. For example, 3 divides 6^2, resulting in 36, implying that 3 divides 6.
The sum or difference of a rational and an irrational number is irrational
A non-zero rational and irrational number’s product and quotient are both irrational.
√p is irrational when p is a prime number. For example, 7 is a prime number and √7 is irrational.
Proof by Contradiction
In the method of contradiction, to check whether a statement is TRUE
(i) We assume that the given statement is TRUE.
(ii) We arrive at some result that contradicts our assumption, thereby proving the contrary.
Eg: Prove that √7 is irrational.
Assumption: √7 is rational.
Since it is rational √7 can be expressed as √7 = a/b, where a and b are co-prime Integers, b ≠ 0. On squaring, a^2/b^2=7 ⇒a^2=7b^2.
Hence, 7 divides a. Then, there exists a number c such that a=7c. Then, a^2=49c^2. Hence, 7b^2=49c^2 or b^2=7c^2.
Hence 7 divides b. Since 7 is a common factor for both a and b, it contradicts our assumption that a and b are coprime integers.
Hence, our initial assumption that √7 is rational is wrong. Therefore, √7 is irrational.
Terminating and nonterminating decimals:
Terminating decimals are decimals that end at a certain point. Example: 0.2, 2.56, and so on.
Non-terminating decimals are decimals where the digits after the decimal point don’t terminate. Example: 0.333333….., 0.13135235343…
Non-terminating decimals can be :
a) Recurring – a part of the decimal repeats indefinitely (0.142857142857….)
b) Non-recurring – no part of the decimal repeats indefinitely. Example: π=3.1415926535…
Check if a given rational number is terminating or not
If a/b is a rational number, then its decimal expansion would terminate if both of the following conditions are satisfied :
a) The H.C.F of a and b is 1.
b) b can be expressed as a prime factorization of 2 and 5 i.e b=2^m×5^n where either m or n, or both can = 0.
If the prime factorization of b contains any number other than 2 or 5, then the decimal expansion of that number will be recurring
Example:
1/40=0.025 is a terminating decimal, as the H.C.F of 1 and 40 is 1, and the denominator (40) can be expressed as 2^3×5^1.
3/7=0.428571 is a recurring decimal as the H.C.F of 3 and 7 is 1 and the denominator (7) is equal to 7^1
Example: Prove that √3 is irrational.
Solution: Let us assume that √3 is rational.
That is, we can find integers a and b (≠ 0) such that 3 =a/b.
Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.
So, b√3=a
Squaring both sides, we get 3b^2= a^2
Therefore, a^2 is divisible by 3, and we know that If p(a prime number) divides a^2 , then p divides a as well.
It means a is also divisible by 3.
So, we can write a = 3c for some integer c.
Substituting for a, we get
3b^2= 9c^2,
that is, b^2= 3c^2
This means that b^2 is divisible by 3, and so b is also divisible by 3.
Therefore, a and b have at least 3 as common factors. But this contradicts the fact that a and b are coprime.
This contradiction has arisen because of our incorrect assumption that 3 is rational.
So, we conclude that √3 is irrational.
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