Natural Numbers: Natural numbers were the first to come. They are denoted by N. These numbers can be counted on fingers. E.g.: 1, 2, 3, 4, 5, 6, 10, 15, 20, 21 etc.
Whole Number: Aryabahatta, famous Mathematician gave ‘0’ to the number system. It is very powerful number. Anything multiplied by 0 becomes 0. This new number 0, when added to the Natural numbers gave a new set of numbers called Whole number.
E.g.: 0, 2, 3 5 etc. It is denoted by W. It has all natural numbers plus 0. Note that Whole number has only positive numbers. All Natural numbers are whole number but the reverse is not true.
Integers: Field of Mathematics advanced & there was a need for Negative numbers as well. If we add negative numbers to the whole number, we get Integers. It is denoted by “Z”. Z came from word Zahlen that means “to count”.
It is used to express temperature, latitude, longitude etc which can have negative values. E.g.: -20oC. All Whole numbers are Integers, but the reverse is not true. Refer the image below for clarity.
Prime Number: Field of Mathematics advanced & there was a need to divide Natural number in two parts based on divisors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. E.g. 3, 7, 11 etc
Composite Number: A natural number greater than 1 that is not a prime number is called a composite number. E.g. 4, 6, 9 etc
Rational Numbers: Field of Mathematics advance further & concept of division came into picture. Numbers that can be represented in the form of p/q where P& Q are Integers & q≠ 0 were called Rational Number. Word Rational number came from Ratio.
It is demoted by Q. Q letter is taken from word Quotient. E.g.: ½, 9/5 etc. There is Infinite Rational Numbers between any 2 Rational Numbers. All Integers are Rational Number, but the reverse is not true.
Irrational Numbers: Field of Mathematics advance further & mathematicians found that there are some numbers that can’t be written in the form of p/q where p& q are integers & q≠0. They call it irrational Numbers. Eg √2, √3
Real number: Both Rational & Irrational Numbers together forms Ream number. It is denoted by R. Evert point on the number line is a Real number. E.g.: √2, -7, 4/9 , 0, 5 etc. All rational numbers are real number. All irrational numbers are real number, but the reverse is not true.
Euclid’s division algorithm deals with divisibility of integers. It says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. It is mainly used to compute the HCF of two positive integers.
The Fundamental Theorem of Arithmetic deals with multiplication of positive integers. Every composite number can be expressed as a product of primes in a unique way.
It is used to prove the irrationality of many of the numbers & to explore when exactly the decimal expansion of a rational number.
Euclid’s Division Lemma:
A lemma is a proven statement used for proving another statement.
Theorem 1: “Given positive integers a & b, there exist unique integers q & r satisfying a = b*q + r, 0 ≤ r < b”.
Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers.
HCF of two positive integers a and b is the largest positive integer d that divides both a and b
Let’s find HCF of the integers 455 and 42.
We start with the larger integer, that is, 455. Then we use Euclid’s lemma to get
Now consider the divisor 42 and the remainder 35, and apply the division lemma to get
Now consider the divisor 35 and the remainder 7, and apply the division lemma to get
Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is 7.
Numerical: Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.
Solution: Let a be any positive integer and b = 2. Then, by Euclid’s algorithm, a = 2q + r, for some integer q ≥ 0, and r = 0 or r = 1, because 0 ≤ r < 2. So, a = 2q or 2q + 1.
If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive odd integer is of the form 2q + 1.
Numerical: A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?
Solution: This can be done by trial and error. But to do it systematically, we find HCF (420, 130). Then this number will give the maximum number of barfis in each stack and the number of stacks will then be the least. The area of the tray that is used up will be the least.
Now, let us use Euclid’s algorithm to find their HCF. We have :
420 = 130 × 3 + 30
130 = 30 × 4 + 10
30 = 10 × 3 + 0
So, the HCF of 420 and 130 is 10.
Therefore, the sweet seller can make stacks of 10 for both kinds of barfi.
Fundamental Theorem of Arithmetic:
Given by given by Carl Friedrich Gauss, it states that every composite number can be written as the product of powers of primes E.g.: 30 = 2* 3* 5
Theorem 2 : Every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
The prime factorization of a natural number is unique, except for the order of its factors.
E.g. 30 = 2* 3* 5 = 2* 5 * 3 = 3 * 2 * 5 = 3 * 5 * 2 = 5 * 3 * 2 = 5 * 2 * 3
Prime factorization of 30 is unique; it has number 2, 3 & 5 ignoring the order.
Numerical: Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4nends with the digit zero.
Solution: If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5. But as per factorization theorem, 4n has the only prime factor 2 , so it is not divisible by 10.
Numerical: Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Solution: We have : 6 = 21 × 31 and 20 = 2 × 2 × 5 = 22 × 51.
Note that HCF (6, 20) = 21 = 2
HCS is Product of the smallest power of each common prime factor in the numbers.
LCM (6, 20) = 22 × 31 × 51 = 60
LCM is Product of the greatest power of each prime factor, involved in the numbers.
A number ‘s’ is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0. E.g.: E.g.: 1.234674322345634232789064590812343…… or ∏, √2 etc,
Theorem 3 : Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
Theorem 4 : √2 is irrational
The sum or difference of a rational and an irrational number is irrational
The product and quotient of a non-zero rational and irrational number is irrational.
Rational Numbers & decimal Expression:
A rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion.
Theorem 5 : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p , q where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers
Theorem 6 : Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates
Theorem 7 : Let x =p/q be a rational number, such that the prime factorization of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).