Finding Squares Easy method...

Squaring of a number is largely used in mathematical calculations. There are so many rules for special cases. But we will discuss a general rule for squaring which is capable of universal application.

From our childhood we all know one common formula.

i.e (a+b)² = a²+2ab+b²

We can calculate squares using above formula. So let us do some examples

24²=???

To calculate squares by using above formula we have to follow some steps.

i.e let us take unit digit as ‘b’ and remaining part as ‘a’.

in 24, a=2 and b=4.

We should start our calculation from the right side of the formula.

Step-1: b²=16

24²=       6 and 1 as reminder.

Step-2: 2ab=2x2x4=16

 24²=       76 and 1 as reminder. (16+1=17)

Step-3: a²=4

24²=   576 (answer).

This model is time taking to explain but it is very easy to calculate. Now we do some exercise.

Calculate squares for the following numbers.

27, 36, 85, 99

Now how are you feeling? Is it hard or easy??

But it has some limits. It is useful up to 99 only. We can calculate squares more than 99 by using this formula but it seems some difficult. So we need to have some easy and it should satisfy all the conditions.

Now we are moving to next method which is very useful to calculate square of any number whether it is 3 digits or 5 digits.

i.e

Duplex Combination Process

Here I’ll explain this method with an example

897²=???

Explanation:

1.    Square of unit digit. i.e 7²=49; writte down 9 and carry over 4.

2.    2x9x7=126+4=130; writte down 0 and carry over 13. (ten digit and unit digit)

3.    2x8x7+9²=112+81=193+13=206; writte down 6 and carry over20. (ten digit square and multiplication of unit digit and hundred digit.

4.    2x8x9=144+20=164; writte down 4 and carry over 16.

5.    8²=64+16=80

So the final answer is 804609.

Lets we do another example

1432²=????  Ans: 2₁0₂5₃0₂6₁24

Explanation:

1.    2²=4; Write it down.

2.    2x3x2=12; write down 2 and carry over 1

3.    2x4x2+3²=25+1=26; write down 6 and carry over 2.

4.    2x1x2+2x4x3=28+2=30; write down 0 and carry over 3.

5.    2x1x3+4²=22+3=25; write down 5 and carry over 2.

6.    2x1x4=8+2=10; write down 0 and carry over 1.

7.    1²+1=2

Final Answer is 2050624

So now Practice some squares…

12345²=??

87532²=??

And do your own numbers…

Some special cases….

A)  Square of a number with unit digit as 5

It is very easy to calculate square of a number with unit digit as 5. Lets take an example.

25²=???

Ans is 625.

Explanation: 5²=25; write down 25

Next number for remaining digit 2 is 3. Multiply 2and 3 we get 6. Now write down.

The final answer is 625…

Lets try following numbers…

35,45,75,95,115

Square of a number which is nearer to 10^x

For this type of numbers we use following formula

(a+b)²=a²+2ab+b²

(a-b)²=a²-2ab+b²

Ex:

1)    98²=??

98=100-2

(100-2)²=10000-400+4=9604

2)    107²=(100+7)²=10000+1400+49=11449

3)    998²=??

4)    10008²=??

Today we end over topic. Tomorrow we discuss about finding cubes…..

Day 2 Chapter 1 TEST OF DIVISIBILITY

Test of Divisibility

1. Divisibility By 2: A number is divisible by 2, If its unit digit is any of 0, 2, 4, 6, 8

Ex: 876456 is divisible by 2, while 987345 is not.

2. Divisibility by 3: A number is divisible by 3, If the sum of its digits is divisible by 3 or If the sum of the digits is equal to 3, 6, 9.

Ex: 592482 is divisible by 3, since the sum of the digits is 3.

3. Divisible by 4: A number is divisible by 4, if the number formed by last two digits of the number is divisible by 4.

Ex: 892648 is divisible by 4, since the last two digits are divisible by 4.

4. Divisible by 5: A number is divisible by 5, if its unit’s digit either 5 or 0.

Ex: 87654560 is divisible by 5, since the last digit is 0.

5. Divisible by 6: A number is divisible by 6, if following conditions satisfies.

a.       The number should be even.

b.      The sum of all digits should be divisible by 3.

Ex: 35256 is divisible by 6, since it is even and the sum is divisible by 3.

6. Divisible by 8: A number is divisible by 8, if the number formed by the last three digits of the number is divisible by 8.

Ex: 953360 is divisible by 8, since last three digits (360) is divisible by 8.

7. Divisible by 9: A number is divisible by 9, if the sum of the digits is equal to 9.

Ex: 99999 is divisible by 9, since the sum of digits is equal to 9.

8. Divisible by 10: A number is divisible by 10, if its unit digit is 0.

Ex: 123450 is divisible by 10, since its unit digit is 0.

9. Divisible by 11: A number is divisible by 11, If the difference of the sum of its digits at odd places and the sum of even places, is either 0 or a number divisible by 11.

Ex: 999999 is divisible by 11, since the difference of the sum of its digits at odd places and the sum of even places is 0.

10. Divisible by 12: A number is divisible by 12, If it divisible by both 3 and 4.

Day 1 Chapter 1 NUMBER SYSTEM-1

For any mathematical calculations, use of a number system is important. In this chapter we present some fundamental properties of number system, which you will require while solving problems in all competitive exams

Important Definitions

Face Value

Face Value of a digit in a number is the value of the digit itself wherever it may be.

Example: in 720, the face value of 2 is 2 only.

Place Value

As the name suggest, place value of a digit in a number is the digit multiplied by the place of the digit.

Ex: in 576, the place value of 5 is 5x100=500, and the place value of 6 is 6x1=6.

Various Types of Numbers

Natural Number:

All counting numbers are called Natural Numbers. It is denoted by ‘N’.

N = {1,2,3,4….}

Whole Numbers:

All counting numbers including zero ‘0’ is called Whole Numbers. This is denoted by ‘Z’.

Z= {0,1,2,3….}

Integers:

Which are zero, the natural numbers, and the negatives of the naturals:

..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ...

Rational Numbers:

Which are in the form of p/q, where p and q are integers and q not equal to 0. It is denoted by ‘Q’.

Ex: 2/3, 15/26…..

Irrational Numbers:
The numbers which are in the form of Non repeating and non terminating decimal  is called Irrational Numbers.

Ex: 0.33333,3.141…..

Real Numbers:

All rational an Irrational numbers are called Real Numbers. It is denoted by ‘R’.

R=2/3, 15/26, 0.33333,3.141......

True or False:

• True or False: An integer is a rational number.

Since any integer can be formatted as a fraction by putting it over 1, then this is true.

• True or False: A rational is an integer.

Not necessarily; ⁴/₁ is an integer, but ²/₃ is not! So this is false.

• True or False: A number is either a rational or an irrational, but not both.

True!  In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or not (so it's a rational); there is no overlap between these two number types!

Classify according to number type; some numbers may be of more than one type.

• 0.45

This is a terminating decimal, so it can be written as a fraction: ⁴⁵/₁₀₀ = ⁹/₂₀. Since this fraction does not reduce to a whole number, then it's not an integer or a natural. And everything is a real, so the answer is: rational, real

• 3.14159265358979323846264338327950288419716939937510...

You probably recognize this as being pi, though this may be more decimal places than you customarily use. The point, however, is that the decimal does not repeat, so pi is an irrational. And everything (that you know about so far) is a real, so the answer is: irrational, real

• 3.14159

Don't let this fool you! Yes, you often use something like this as an approximation of pi, but it isn't pi! This is a rounded decimal approximation, and, since this approximation terminates, this is actually a rational, unlike pi which is irrational! The answer is: rational, real

• 10

Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text and teacher (there is some inconsistency), this may also be counted as a rational, which technically-speaking it is. And of course it's also a real. The answer is: natural, whole, integer, rational (possibly), real

• ⁵/₃

This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real

• 1 ²/₃

This can also be written as ⁵/₃, which is the same as the previous problem. The answer is:rational, real

• –sqrt(81)

Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies: –sqrt(81) = –9, which is just an integer. The answer is: integer, rational, real

• ^{– 9}/₃

This is a fraction, but notice that it reduces to –3, so this may also count as an integer. The answer is: integer (possibly), rational, real

Even Numbers:

The numbers which are divisible by 2 are called Even numbers.

Ex: 4, 64, 56, 199998….

Odd numbers

The numbers which are not divisible by 2 are called Odd Numbers.

Ex: 3, 57, 444445….

Prime Numbers

The numbers which is divisible by 1 and itself is called Prime Numbers.

Ex: 3,5,7….

Some Important Points:

• Odd x Odd = Odd
• Even x Odd = Even
• Even x Even = Even
• Odd + Odd = Even
• Odd + Even = Odd
• Even + Even = Even