Made by- Vrinda Batra, Ishita Sharma,Shivam Das Gupta.

●NATURAL NUMBERS●WHOLE NUMBERS●INTEGERS●REAL NUMBERS●PRIME FACTORISATION●LEAST COMMON MULTIPLE●HIGHEST COMMON FACTOR●FRACTIONS ●DECIMALS ●RATIONAL NUMBERS●IRRATIONAL NUMBERS●RATIONALISATION●SQUARES AND SQUARE ROOTS●CUBE AND CUBE ROOTS●DIVISIBILITY RULES●FACTS●END

INDEX

●Natural numbers are numbers which are 1 onwards.●Number “0” is not a natural number.●Natural numbers are denoted by the letter “N”.

NATURAL NUMBERS

●Whole numbers are numbers which are 0 onwards.●They are denoted by the letter “W”.●Types of whole numbers :1.Even number- A whole number which is divisible by

2.2.Odd number- A whole number which is not divisible

by 2.3.Prime number- A natural number greater than 1 and

having no factor other than 1 and itself.4.Composite number- A natural number having atleast

one more factor other than 1 and itself.5.Co prime numbers- Two natural numbers having no

common factor other than 1.

WHOLE NUMBERS

●All positive and negative numbers including zero are called integers.●They are denoted by the letter “I” or “Z”.

INTEGERS

They include all of the measuring numbers.They are usually written using decimal numerals. Example- 123.456They are of 2 types –Rational numbersIrrational numbers

REAL NUMBERS

PRIME FACTORISATION ● Prime factorisation is when a natural number is

expressed as the product of prime numbers.● Method: example:4725 3 4725 3 1575 3 525 5 175 5 35 7

Hence: 4725= 3*3*3*5*5*7

LEAST COMMON MULTIPLE

● It is the smallest natural number which is a multiple of both the numbers.

● Method: example:72, 240, 196

2 72, 240, 196 2 36, 120, 98 2 18, 60, 49 3 9, 30, 49 3 3, 10, 49Hence, LCM of 72, 240, 196 is 2*2*2*3*3*10*49 = 35280

HIGHEST COMMON FACTOR● HCF of two natural numbers is the largest common factor

of the given natural numbers.● Method:

example: 72, 126, 270 72= 2*2*2*3*3 126= 2*3*3*7 270= 2*3*3*3*5 Now, we find the common numbers and multiply them: 2*3*3= 18

Hence, HCF= 18

FRACTIONS● A fraction is a number which represents a part of a whole.● Fraction is p/q where p, q are whole numbers and q is not a

0.● The upper part is the numerator while the lower part is the

denominator.● When the numerator is bigger than the denominator, it is

called an improper fraction and when the numerator is smaller than the denominator, then is it called a proper fraction. 5/2, 2/3 are examples of improper and proper fractions respectively.

● When there is a number in front of a proper fraction, then it is called a mixed fraction.ex: 9 2/3.

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● Addition of fractions: 2/3 + 4/6LCM of denominators= LCM of 3 and 6= 6We should match the denominators. To do that we multiply 2/3 and 2 so that we can get the denominator 6. We should do the same for 4/6.

2/3*2/2 + 4/6*1/1= 4/6 + 4/6= 8/6● Subtraction of fractions:

We do the same as we do for addition, but in the last step, instead of adding we subtract.

DECIMALS● Decimals are improper fractions which is expressed as a

number. ex: 1/2 is expressed as 0.5.● Decimals can be converted into fractions by putting zeros

equal to the number of decimal places.ex: 23.8= 238/10. The .8 is representing 1/10 so, we put 10 in the denominator.

● Addition of decimals: To add 23.4 and 56.7 we add 234 and 567. The answer is 801. The highest decimal place is 1/10 so we add a point between 0 and 1. Hence, the answer is 80.1.

● Subtraction of decimals:We use the same method for subtracting, but we subtract instead of adding.

●Rational numbers are like a/b where b is not a zero.●Examples- ¼, ½, ¾ etc.●The symbol for rational numbers is “Q” (for quotient)

RATIONAL NUMBERS

●These are square roots of non square numbers.●They are neither terminating nor recurring decimals.●22/7 is an exception and is an irrational number.●Other examples include- √2 = 1.41421356237….

IRRATIONAL NUMBERS

RATIONALISATION

● If the product of 2 irrational numbers is a rational number, then each number is called the rationalising factor of the other number.

ex:1. 5* 5 = 5, hence sq. root 5 is the rationalising factor.2. ( 5+ 7) (5- 7)= 25-7= 18

SQUARES AND SQUARE ROOTS

● Squares: When a number is multiplied by itself, the number obtained is its square.ex: 9*9= 92 = 81

● Square roots:When we sq. a number, the sq. root of the number obtained is the number. ex: 9 is the sq. root of 81.

● Method: sq. root of 81. 3 81 3 27 3 9 3 3 181= 32*32= 3*3= 9Hence, sq.root of 81 is 9.

CUBE AND CUBE ROOTS● Cube: When a number is multiplied thrice, the number

obtained is its cube.ex: 2*2*2= 8.● Cube root: When we cube a number, the cube root of the

number obtained is the number.ex: cube root of 8= 2● Method: cube root of 729

3 729 3 243 3 81 3 9 3 3 1729= 33*33= 3*3= 9Hence,cube root of 729 is 9.

DIVISIBILITY RULES● Some divisibility rules are:1. 2= The last digit should be even.2. 3= Sum up all the numbers, if the sum is divisible by 3 then

the number is divisible by 3.3. 5= The last digit should be 0 or 5.4. 6= The number should be divisible by 2 and 3.5. 7= The last digit should be multiplied by 2 and should be

subtracted by the rest of the number. Keep doing this till we obtain 7 or 0.ex: 483= 48-(3*2)= 42, 4-(2*2)= 0

6. 11= Subtract the last digit from the rest. Keep doing this till we get 0.

7. 13= Add 4 times the last digit to the rest.ex: 637: 63+ 7*4= 91= 9+ 1*4=13. Hence, 637 is divisible by 13.

●They were the first to develop a base ten system. They developed methods of expressing every possible number using a set of ten symbols.●Zero was used to denote an empty space.●The square of an odd natural number is an odd natural number.

FACTS