NUMBER

SYSTEM

-3

-1

-2

320 1

NUMBER LINE

1/2

-1/2

A number line consist of all types :• Natural numbers• Whole numbers• Integers• Rational numbers • Irrational numbers

CLASSIFICATION OF NUMBERS

NATURAL NUMBERS

Counting numbers are called Natural numbers. 1 is the first natural number and there is no last natural number. The collection of all natural numbers is denoted by N.

321 4 5 76 oo

WHOLE NUMBERS

210 3 4 65 oo

0 is the first whole number and there is no last whole number. The collection of all whole numbers is denoted by W.

INTEGERS

-1

-2

-3

0 1 32 oo

When we include negative numbers like -1,-2,-3,-4 etc in the collection of whole numbers then we will get the collection of integers. It is denoted by Z.

-00

RATIONAL NUMBERS

-2/2-4/2-6/2 0 2/2 6/24/2 00

A number r is called a rational number. If it can be written in the form p/q, where p and q are integers and q is not equal to 0. The collection of rational numbers is denoted by Q.

-00 -1/2 1/2

IRRATIONAL NUMBERSA number r is called a irrational number. If it cannot be written in the form p/q, where p and q are integers and q is not equal to 0.

√14√13 √3

√11√5

√2

√12

√6√10 √7

√17

√15

√19√18

√20

REPRESENTING IRRATIONAL

NUMBERS ON NUMBER LINE

2BO 3 4 65

1. Firstly draw a number line.2. Then take OB as the base.

3. Then make a line AB with measurement of 1 cm. 4. Then join A and O.

5. Finally use a compass with centre O and radius OB, draw an arc intersecting the number line at the point P.

LOCATE √2 ON NUMBER LINE

P

A √2

DECIMAL EXPANSION

Now we will look at the decimal expansion of rational and irrational numbers to distinguish between them.Try the decimal expansion of the following numbers 10/3 7/8 √5

10/3 = 3.3333… 7/8 = 0.875 √5 = 2.23606797749979

We have noticed that :- 1. The remainders either became 0, or start repeating themselves.2. The number of entries in the repeating string of remainders is

less than the divisor.3. If the remainders repeat, then we get a repeating block of

digits 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.

CONVERSION OF DECIMAL

NUMBERS INTO RATIONAL

NUMBERS OF THE FORM P/Q

CONVERSION OF A TERMINATING DECIMAL NUMBER TO THE FORM P/Q

In order to convert a rational number having finite number of digits after the decimal point, we follow the following steps:-STEP 1- Obtain the rational number.STEP 2- Determine the number of digits in its decimal part.STEP 3- Remove decimal point from the numerator. Write 1 in the denominator and put as many zeros on the right side of 1 as the number of digits in the decimal part of the given rational number.STEP 4- Find a common divisor of the numerator and denominator and express the rational number to lowest terms by dividing its numerator and denominator by the common divisor.

CONVERSION OF A TERMINATING DECIMAL NUMBER TO THE FORM P/Q

Example – Express the following decimal in the form p/q.a) 0.15 b) 0.675Solution:-b) Let x= 0.15 (i)Multiplying (i) by 100100x = 15(ii)0.15 = 15/100 = 3/20b) Let x=0.675 (i)Multiplying (i) by 10001000x = 675 (ii)0.675 = 675/1000 = 27/40

CONVERSION OF A PURE RECURRING DECIMAL TO THE FORM P/Q

In order to convert a pure recurring decimal to the form p/q, we follow the following steps:-STEP 1- Obtain the repeating decimal and put it equal to x.STEP 2- Write the number in decimal form by removing bar from the top of repeating digit and listing repeating digit at least twice.STEP 3- Determine the number of digits having bar on their heads.STEP 4- If the repeating decimal has one place repetition, multiply by ten; a two place repetition, multiply by hundred; a three place repetition, multiply by thousand and so on. STEP 5- Subtract the number in step two from the number obtained in step four.

CONVERSION OF A PURE RECURRING DECIMAL TO THE FORM P/Q

STEP6- Divide both sides of the equation by the coefficient of x. STEP 7- Write the rational number in its simplest form. Example – Express the following decimal in the form p/q.a) 0.111… b) 0.66666…..Solution:-b) Let x= 0.111… (i)Multiplying (i) by 1010x = 1.111…(ii)Subtracting (i) by (ii)9x = 10.111… = 1/9 b) Let x=0.666…(i)Multiplying (i) by 1010x = 6.666… (ii)Subtracting (i) by (ii)9x = 60.666… = 2/3

In order to convert a mixed recurring decimal to the form p/q, we follow the following steps:-STEP 1- Obtain the mixed recurring decimal and put it equal to x.STEP 2- Determine the number of digits after the decimal point which do not bar on them. Let there be n digits without bar just after the decimal point.STEP 3- Multiply both sides of x by 10n so that only the repeating decimal is on the right side of the decimal point.STEP 4- Use the method of converting pure recurring decimal to the form p/q and obtain the value of x.

CONVERSION OF A MIXED RECURRING DECIMAL TO THE FORM P/Q

CONVERSION OF A MIXED RECURRING DECIMAL TO THE FORM P/Q

Example – Express the following decimal in the form p/q.a) 0.3222… b) 0.12333……Solution:-b) Let x= 0.3222… (i)Multiplying (i) by 1010x = 3.222…(ii)Multiplying (ii) by 10100x = 32.222…(iii)Subtracting (ii) by (iii)90x = 290.3222… = 29/90. b) Let x=0.12333… (i)Multiplying (i) by 100100x =12.333… (ii)Multiplying (ii) by 101000x = 123.333… (iii) Subtracting (ii) by (iii)900x = 1110.12333… = 111/900

GEOMETRICALLY REPRESENTATION OF IRRATIONAL

NUMBERS

GEOMETRICALLY REPRESENTATION OF

IRRATIONAL NUMBERSIn order to represent a irrational number geometrically, we follow the following steps:-STEP 1- Obtain the positive real number x.STEP 2- Draw a line and mark a point A on it.STEP 3- Mark a point B on the line such that AB=x units.STEP 4- From point b mark a distance of 1 unit and mark a new point as C.STEP 5 – Find the mid-point of AC and mark the point as O. STEP 6 – Draw a circle with centre O and radius OC.STEP 7 – Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Length BD is equal to √x.

THANK YOU