|Class 10 Maths Real Numbers||Real Number System|
Real Number System
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.