Math : Shapes

Area

The amount of space inside the boundary of a flat object that is 
2 dimensional objects such as square,triangle,circle.

Perimeter

The perimeter is the distance around the 2 dimensional object.In circle, perimeter is referred as circumference.

Volume

Volume is nothing but the Capacity. The amount of space an object occupies 
(3D).

Surface area

The total area of the surface of a 3D object.

Shapes

Regular polygon

 A polygon is regular when all sides are equal and all angles are equal. If its not regular then we can say its irregular polygon

Quadrilateral

A quadrilateral has four sides which are closed and it is 2D. If we add its interior angle then it should be 360°.

Math : Roots

Square root

The square root of a number is a value that when multiplied by itself gives the number.

Example 1

2*2 = 4, here to get the square root of 4 we multiplied 2 by itself which gives 4.
Hence the square root of 4 is 2
That is, √4 = 2

Example 2

3*3 = 9, here to get the square root of 9 we multiplied 3 by itself which gives 9.
So the square root of 9 is 3
That is, √9 = 3

Cube root

The cube root of a number is a value that when multiplied by itself 3 times gives the number

Example

2*2*2 = 8, here to get the cube root of 8 we multiplied 2 by itself 3 times gives 8.
Hence the cube root of 8 = 2

Math : 0/0,1/0

1/0 is undefined

Let us assume
1/0.1 = 10
1/0.01 = 100
1/0.0001 = 1000
.
.
.
So the value keeps on increasing
In this case we can say 1/0 = infinity (endless)

Let's assume negative value

1/-0.1 =-10
1/-0.01 = -100
.
.
.
So in this we can say 1/0 = -infinity

We cannot able to define the value

Hence 1/0 is undefined.


0/0 is indeterminate ( could be any value )

Let's assume the closer value of 0

0.1/0.1 = 1
0.01/0.01 = 1

In this case we gets 0/0=1

Now let's keep the numerator as 0 and assume denominator value closer to 0

0/0.1=0
0/0.01=0

In this case we gets 0/0=0

So 0/0 be any value that is 0 or 1

Hence 0/0 is indeterminate

Math : Powers

Power

The power of a number says how many times  to use the number in multiplication.

Powers are also called Exponents or Indices ( index ).

Example 1

Find 5 to the power 2 or 5 to the second power or 5 squared

5^2 = 5 * 5 = 25

Example 2

Find 5 to the power 3 or 5 to the third power or 5 cube

5^3 = 5 * 5 * 5 = 125

Example 3

Find 5 to the power 4

5^4=5*5*5*5 = 625

Negative Exponent

Negative exponent which means how many times to divide one by that number

Example

2^(-3) = 1/(2*2*2) = 1/8

3^(-2) = 1/(3*3) = 1/9

We can say that 2^(-3) = 1/(2^3)

Note 1: If the exponent is 1, that is any number have the exponent 1 then the solution be the number itself.

5^1 = 5 , 8^1 = 8 , (-3)^1 = -3

Note 2 : if the exponent is 0, that is any number which has the exponent as 0 then the solution be 1.

5^0 = 1

That is,
  5^3=5*5*5=125
  5^2=5*5=25
  5^1=5
5^0= ?
5^-1=1/5
5^-2=1/5*5=1/25

In the above example, the exponent keeps on reducing by 1 and to the right side the number gets divisible by 5

When 5/5 we gets 1

Hence 5^0=1

Note 3 : If whole number 0 has exponent 0 that is 0^0 which gives 1

Let's consider the closest value of 0,
 0.1^0.1 = 0.7943282
0.01^0.01=0.954992586
0.001^0.001=0.993116

So as we approach to 0 we gets 1
That is, limit x->0, x^x=1

Laws of Exponent

 1. x^0 = 1

 2. x^1 = x

 3. x^-1 = 1/x  [ x^-m = 1/x^m]

 4. x^m * x^n = x^(m+n)

 5. x^m/x^n = x^(m-n)

 6. (x^m)^n = x^(m*n)

 7. (x*y)^m = x^m * y^n

 8. (x/y)^m = x^m/y^m

Math : Sieve of Eratosthenes - Finding Prime Numbers

Finding prime numbers

 We can find the prime numbers by using Sieve of Eratosthenes method. (Eratosthenes is a great mathematician who created an efficient method for finding prime numbers)

Let's learn method through example

Example

Find all prime numbers up to 20

 Step 1

Write all the numbers till 15

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15

As we know 1 is not a prime number so strike out 1

Step 2

2,3,4,5,6,7,8,9,10,11,12,13,14,15

Here 2 is a prime number and strike out all the numbers which are divisible by 2
(4,6,8,10,12,14 are divisible by 2, so remove all those numbers)
2,3,5,7,9,11,13,15

Step 3

 we should take the next number 3 and proceed like step 2 (9,15 are dvisible by 3, so remove those numbers)
2,3,5,7,11,13

Similarly we should keep going

For our example,
2,3,5,7,11,13 are the prime numbers till 15

Math : Prime Factorization

Factors

 Factors are the number we multiply together to get another number.

Example

 3*4=12

3 and 4 are the factors of 12

Prime number

 A whole number which has exactly two factor that is 1 and the number itself.

2,3,5,7...are some of the prime numbers.

Prime factorization

 A prime factorization is finding which prime numbers multiply together to get the original number

Example

12 = 4*3=6*2=2*3*2

In the above example, 4 and 6 are not prime numbers.
In, 12=2*3*2 where 2,3,2 are prime numbers.
Hence the prime factors of 12 is 2 , 3 , 2.

Math : LCM & GCF

Greatest Common Factor (GCF)

 The largest common factor of two or more groups. GCF is also known as Greatest common divisor ( GCD ) or Highest Common Factor (HCF )

Example 1

8 = 1 , 2 , 4 , 8
6 = 1 , 2 , 3 , 6

Here, 1,2 are the two common factors of 8 and 6. The largest is 2 hence 2 is the GCF of 8 and 6.

Example 2

10 = 1 , 2 , 5 , 10
20 = 1 , 2 , 4 , 5 , 10 , 20

Here 1,2,5,10 are the common factors of 10 and 20.
10 is the largest common factor. Hence 10 is the GCF of 10 and 20.

Least Common Multiple

 The smallest positive number that is the multiple of two or more numbers.

Example 1

3 = 3,6,9,12,15,18,21,24,27...
4 = 4,8,12,16,20,24,28...
Here 12,24... Are the common multiples of 3 and 4.
12 is the least common multiple.

Example 2

2 = 2,4,6,8,10,12,14,16..
3 = 3,6,9,12,15,18,21...
Here 6,12,18...are the common multiples of 2 and 3.
6 is the least common multiple.

Note : LCD is least common denominator.

Math : Multiples

Multiples

 A multiple is just simply multiplying a number by an INTEGER ( not fraction or decimal ).

Example 1

Multiples of 2

 ... -6,-4,-2,0,2,4,6,8...

Note :  10 is a multiple of 2 since 2*5=10, but 9 is not a multiple of 2.

Example 2

Multiples of 3

...-9,-6,-3,0,3,6,9...

Factors

Factors

 Factors which means the numbers we can multiply together to get another number.

Example 1

   2 * 5 = 10

Here, 2 and 5 are the factors of 10.

Example 2

 4 * 6 = 24

Here 4 and 6 are the factors of 24

Also,

 2 * 12 = 24

Here 2 and 12 are the factors of 24.

Note : A number can have many factors.

Math : Prime and Composite numbers

Prime Numbers

 A prime number is a whole number which has ONLY 2 Factors.

Example

Let us consider a whole number 2
2 = 1*2 = 2*1
Here, 1 and 2 are the only 2 factors for 2.

Now let we check 3
3 = 1*3 = 3*1
Here, 1 and 3 are the only 3 factors for 3.

So the prime numbers are 2,3,5,7,11,13...

Note: Prime numbers which can be divided evenly by 1 and by the number itself.

Composite numbers

 A composite number is a whole number which has more than 2 factors.

Example

4 = 1*4 = 2*2 = 4*1

Here, 1,2,4 are the factors for 4. A whole number 4 has 3 factors. Hence 4 is a composite numbers.

4,6,8,9,10,12... are composite numbers.

ONE

1 is neither a prime nor a composite number.
1 = 1*1
1 has only one factor. Hence 1 is not a prime and not a composite number.

Even and odd numbers

Even numbers

 Any number that can be shared evenly for two groups without breaking it, is called an even numbers.

Example
 I have 4 apples and I wants to split equally and give this to A and B. 2 apples for A and 2 apples for B. Here I splited and gave equal number of apples to each and no apples remaining.

 Mathematically, any integer that can be divided by 2 is an even number.
2,4,6,8,10,12...  are even numbers.

Odd numbers

 Any number that cannot be shared evenly for two groups is called an odd number.

Example
 I have 5 apples and I give 2apples to A and 2apples to B equally, but 1 more apple remains.
 Mathematically, any integer which cannot be divided by 2 is odd number.
1,3,5,7,9,11... are odd numbers

Addition

Odd number + Odd number = Even number (3+7=10)
Even number + Even number = Even number (4+2=6)
Odd number + Even number = Odd number (3+4=7)
Even number + odd number = Odd number (6+1=7)

Multiplication

Odd number * Odd number = Odd number (5*3=15)
Even number * Even number = Even number (6*8=48)
Odd number * Even number = Even number (5*10=50)
Even number * Odd number = Even number (4*7=28)

Zero

Let us assume 0 be an odd number,
 Odd + odd = even
Here, 0 + 0 = 0 ( but we considered 0 be an odd number)
Hence in this case we cannot able to say odd + odd = even

So 0 is not an odd number

Now, let us assume 0 be an even number

Odd + even = odd
3 + 0 = 3

Similarly, it will holds good for all other laws.

Hence 0 be an Even number.

Mathematics : Numbers

Natural numbers

 A natural number is a number that occurs commonly in  nature.
That is we naturally use numbers to count 1,2,3,4,5...

Whole numbers

 Whole numbers are simply the numbers from 0,1,2,3,4...

Note : Natural numbers can also be referred as whole numbers but whole numbers which cannot be a natural number.
That is 0 cannot be a natural number.

Integers

 An integers is a positive and negative whole numbers.
That is ...-3,-2,-1,0,1,2,3...

Fractions

In the above figure, an apple is divided EQUALLY into 8 pieces. Out of 8 pieces, let me eat 2 pieces. That is I have eaten 2/8, remaining 6/8 left out. That is out of 8 pieces, 6 pieces remains.
2/8, 6/8 are fractions.
The top number which is called as NUMERATOR.
The bottom number is called as DENOMINATOR.
Mathematically, any number which can be written in the form of p/q where p and q are integers and q not equal to 0.

Decimal Number

 A decimal number is a number that uses a decimal point followed by digits that show the value smaller than 1.
After decimal point the digits takes the place values like 1/10 , 1/100, 1/1000...
that is tenths, hundredths, thousandths... 

Example

23.7 is a decimal number. 
That is 20+3+7/10.

What is infinity?
 Infinity is something which has NO END. Infinity is not a big number, not extremely huge number its endless.

What is an imaginary number?
 Usually when we square a number we gets positive number or 0.
But imaginary number is, when squared it gives a negative result.
i=sqrt(-1).

Real Numbers Other than infinity and imaginary number, all other numbers are called as Real numbers.

Rational Numbers

 Any number which can be written in the form of ratio is called as Rational number. That is a number that can be made by diving two integers.

Example

* Natural number : 10 which can also be written as 10/1 which gives 10
* Whole number : 0 = 0/1=0
* Fraction : 4/6
* Decimal : 0.5 = 1/2
* Integer : -4/2
* special case : 7.77777... ( it doesn't have an end, but we know its end number be 7)

Irrational numbers A real number which can not be written in the ratio form.

Example

pi=3.141592... ( it doesn't have end ), so we can not write this as a ratio.