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➣ Counting numbers - 1, 2, 3,...
➣ Zero - 0
➣ Negative numbers -...,-3,-2,-1
➣ Fractions - Number representing part of a whole.
➣ Decimals - Numbers representing both integral and fractional part .
Different sets found to group these numbers
➢ Natural Numbers, N = {Counting numbers}
➢ Whole numbers, W = {0,N}
➢ Integers, Z = {...,3,2,1,0,1,2,3,...}
➢ Rational Numbers, Q = {p/q, q≠0},
→ Every integers can be expressed as p/q with q=1.
→ The rational numbers do not have a unique representation in the form p/q , where p and q are integers and q ≠ 0. For example, 1/2= 2/4 =10/20 =25/50 = 47/94, and so on. These are equivalent rational numbers (or fractions).
→ On the number line, among the infinitely many fractions equivalent to 1/2, we will choose 1/2 i.e., the simplest form to represent all of them.
→ There are infinitely many rational numbers between any two given rational numbers.
Is the rational number the complete set of numbers ?
Are there numbers which can't be represented as rational number ?
↬ Rational number is not a complete set of numbers.
↬ There are infinitely many numbers that can not be represented as a ratio of two integers.These numbers are called Irrational numbers.
e.g. √2, 𝜋 (22/7 is its approximate value) etc.
"An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor reoccurring."
➧ In 425 BC,Theodorus of Cyrene showed that √3, √5, √6, √7 , √10 , √11, √12, √13, √14, √15 and √17 are also irrationals.
➧ π was known to various cultures for thousands of years, it was proved to be irrational by Lambert and Legendre only in the late 1700s.
➧ Square root of Prime numbers are irrational.
Irrational numbers on the number line
Irrational numbers like √2, √3, √5 etc can be shown on number line using Pythagoras theorem.
√2 on number line
→ First, we consider a unit square OABC, with each side 1 unit in length. Then we can see by the Pythagoras theorem that, length of the diagonalOB = √(12 + 12) = √2 .
→ We transfer this square onto the number line making sure that the vertex O coincides with zero.
→ Using a compass with centre O and radius OB, draw an arc intersecting the number line at the point P. Then P corresponds to √2 on the number line.
√3 on number line
→ Construct BD of unit length perpendicular to OB.
→ Then using the Pythagoras theorem, we see that OD = √(√2)2 + 12 = √3 .
→ Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the point Q. Then Q corresponds to √3 .
➧ In the same way, we can locate √n for any positive integer n, after √(n − 1) has been located.
Difference Between Rational & Irrational numbers
Decimal expansions of rationals and irrationals numbers can be used to distinguish between them.Decimal expansions of Rational Numbers
On division of p by q, two main things happen – either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.➢ When the remainder becomes zero, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating.
e.g, 3/4 = 0.75, 7/8 = 0.875.
➢ When the remainder never becomes zero, we get a repeating string of remainders and we have a repeating block of digits in the quotient. We say that this expansion is non-terminating recurring.
Such numbers are represented by bar above the block of digits that repeats.
e.g, 4/3 = 1.333... = 1.3, 1/7 = 0.142857142857...= 0.142857
The number of entries in the repeating string of quotient is generally less than the divisor.
➢ How can we express terminating & non-terminating reoccurring decimals in the form p/q, where p and q are integers and q ≠ 0.
→ Terminating decimals can be expressed in the form of p/q by first putting 1 in denominator and adding same number of zeros after it as number of digits in numerator after the decimal point and then removing the decimal point.
2.1423 = 21423/10000
→ Non-terminating reoccurring decimals can be expressed in the form of p/q by using algebraic equations.
e.g., Expressing 0.7777...= 0.7 in the form p/q; where p & q are integers & q ≠ 0.
Since we do not know what 0.7 is , let us call it ‘x’
so x = 0.7777... (i)
Now here is where the trick comes in.
Multiplying both side with 10, we get
10 x = 10 × (0.777...) = 7.777... (ii)
Subtracting (i) from (ii), we get
10x - x = 7.777... - 0.777...
⇒ 9x = 7
⇒ x = 7/9
∴ 0.7= 7/9
Decimal expansions of an Irrational number is non-terminating non-recurring.
→ √2 = 1.4142135623730950488016887242096...→ π = 3.14159265358979323846264338327950...
→ We often take 22/7 as an approximate value for π, but π ≠ 22/7.
→ The Greek genius Archimedes (287 BCE – 212 BCE) was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 – 550 C.E.), the great Indian mathematician and astronomer, found the value of π correct to four decimal places (3.1416). Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places!
The decimal expansion of a rational number is either terminating or non-terminating recurring whereas the decimal expansion of an irrational number is non-terminating non-recurring.
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