Number System; Rational & Irrational Numbers

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A Mathematical and Good Looking Power Point Presentation on "Number System". Ideal for school projects, domestic use and personal education. Contains useful information on the topic "Number System" along with information on great mathematicians.

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Math's Holidays Home Work :

Math's Holidays Home Work By – Ashutosh Singh F rom – Class IX ‘D’

Number System:

Introduction Denotation of Symbols in Number System Natural Numbers N Whole Numbers W Integers Z Rational Numbers Q Real numbers R Number System

IRRATIONAL NUMBERS:

The numbers which cannot be expressed in the form of where p and q are integers are known as Irrational Numbers. SOME EXAMPLES ARE:- , , 0.1011011101111011111… Pythagoras Property :- In a right angled triangle, sum of square of two sides is always equal to the square of third side(hypotenuse).   IRRATIONAL NUMBERS

Pythagoras Pythagoras (582-500bc), Greek philosopher and mathematician, whose doctrines strongly influenced Plato. Born on the island of Sámos, Pythagoras was instructed in the teachings of the early Ionian philosophers Thales, Anaximander, and Anaximenes. Pythagoras is said to have been driven from Sámos by his disgust for the tyranny of Polycrates. About 530 bc Pythagoras settled in Crotona, a Greek colony in southern Italy, where he founded a movement with religious, political, and philosophical aims, known as Pythagoreanism. The philosophy of Pythagoras is known only through the work of his disciples. :

Pythagoras Pythagoras  ( 582-500 bc ), Greek philosopher and mathematician , whose doctrines strongly influenced Plato. Born  on the island of Sámos, Pythagoras was instructed in the teachings of the early Ionian philosophers Thales, Anaximander, and Anaximenes. Pythagoras is said to have been driven from Sámos by his disgust for the tyranny of Polycrates. About 530 bc Pythagoras settled in Crotona, a Greek colony in southern Italy, where he founded a movement with religious, political, and philosophical aims, known as Pythagoreanism. The philosophy of Pythagoras is known only through the work of his disciples.

Decimal Expansion:

Decimal Expansion Now we will look at the decimal expansion of rational and irrational numbers to distinguish in between them. Try the decimal expansion of the following numbers. a.)  

a.) 10/3 = 3.333… b.) 7/8 = 0.875 c.)√5 = 2.23606797749… We have noticed that:- 1.) The reminder either becomes zero or starts repeating themselves. 2.) The number of entries in the repeating string of remainder is less than the divisor. 3.) If the remainder repeats, then we get a repeating block of digit in the quotient.:

a.) = 3.333… b.) = 0.875 c.) = 2.23606797749… We have noticed that:- 1.) The reminder either becomes zero or starts repeating themselves. 2.) The number of entries in the repeating string of remainder is less than the divisor. 3.) If the remainder repeats, then we get a repeating block of digit in the quotient.  

Note:- In case (i) the remainder never becomes zero and repeats after a certain stage forcing the decimal expansion to go for ever. These type of decimal expansions are known as non-terminating repeating decimal expansion. In case (ii) the remainder becomes zero after a certain stage. This type of decimal expansion is commonly known as terminating decimal expansion. In case (iii) the reminder never becomes zero and never repeats. These type of decimal expansion are called non-terminating non-repeating decimal expansion.:

Note:- In case (i) the remainder never becomes zero and repeats after a certain stage forcing the decimal expansion to go for ever. These type of decimal expansions are known as non-terminating repeating decimal expansion. In case (ii) the remainder becomes zero after a certain stage. This type of decimal expansion is commonly known as terminating decimal expansion. In case (iii) the reminder never becomes zero and never repeats. These type of decimal expansion are called non-terminating non-repeating decimal expansion.

Rational Numbers in the form p/( q):

Rational Numbers in the form   Example:- Show that 3.154795462 is a rational number Solution:- We have 3.154795462 = , & hence it is a rational number. Now we will consider the case when the decimal expansion is a non-terminating recurring number. Example:- Show that 0. is a rational number. Solution:- Since we do not know what 0. is, let us call it ‘ x ’ and so x = 0.3333… Now here is where the trick comes in. Multiplying both side by 10 10 x = 10 (0.3333…) = 3.3333… Now, 3.3333… = 3 + x , since x = 0.3333... Therefore, 10x = 3 + x Solving for x , we get 9 x = 3, i.e., x =  

Archimedes:

Archimedes (287-212 bc ), preeminent Greek mathematician and inventor, who wrote important works on plane and solid geometry, arithmetic, and mechanics. Archimedes was born in Syracuse, Sicily, and educated in Alexandria, Egypt. In pure mathematics he anticipated many of the discoveries of modern science, such as the integral calculus, through his studies of the areas and volumes of curved solid figures and the areas of plane figures. He also proved that the volume of a sphere is two-thirds the volume of a cylinder that circumscribes the sphere. In mechanics, Archimedes defined the principle of the lever and is credited with inventing the compound pulley. During his stay in Egypt he invented the hydraulic screw for raising water from a lower to a higher level. He is best known for discovering the law of hydrostatics, often called Archimedes' principle, which states that a body immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. This discovery is said to have been made as Archimedes stepped into his bath and perceived the displaced water overflowing . Archimedes

Irrational Numbers Between Rational Numbers:

F inding irrational numbers between rational numbers is very easy. Let us illustrate it with the help of an example. Example: - Find five irrational numbers between and Solution: - We saw that = 0.142857. So, you can easily calculate = 0.285714. To find an irrational number between and , we find a number which is non-terminating non- repeating lying between them. Of course, you can find infinitely many such numbers. Some of these are:- i) 0.150150015000150000… iii) 0.120120012000120000… ii) 0.03033033303333… iv) 0.110110011000110000… v) 0.140140014000140000…   Irrational Numbers Between Rational Numbers

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