Number systems-II

The concept of fractions, though a simple one, can be often confused. Having not solved questions based on this simple concept, students often tend to confuse the problems. These questions can throw up the occasional challenge and it makes sense to practice these questions from this area.
Definition : Technically, fraction is defined as part of the whole. The most common example of a fraction that comes to mind is half. When we say give me half of something, we are essentially demanding ½ part of it, in others word, ½ is the fractional representation for half.
Fractions are nothing else than the numerator divided by denominator, that is they occur in the form X/Y where X is the numerator and y is the denominator.
Remember : The numerator represents how many parts of that whole are being considered. To remember simply, numerator is the top number of the fraction that represents the numbers of that that are to be chosen. The denominator represents the total number of parts created from the whole, in other words it is the bottom number representing the total number of parts created.
Example of Fractions : ½,2/3,3/4, and the numbers which are in the form of x/y
Types of Fractions:
Proper Fraction: When Numerator < Denominator, then the fraction is called as proper fraction. For example: 2/3, 4/5, 6/7 etc.
Improper fractions: When Numerator > Denominator, then the fraction is called as improper fraction. For example: 5/3, 7/5, 19/7 etc.
Mixed Numbers: when a natural number combines with a fraction that is called a mixed number. For Example: 2¹/₂ ,3⁴/₅  etc
Tool tip 1: Basic Applications of Fractions 1. Fraction help us determine the part of any number ¾ part of 56 = ¾ x 56 = 42, 4/5 part of 90 = 4/5 x 90 =24
2. You can be asked to represent a number in the form of fraction. For example, you can be asked to represent 15 as a fraction of 450. This can be done as follows: 15/450 = 1/30
We have solved the above example and it can be easily seen that 15 is our numerator and 450 is our denominator
3. Always remember that the major quantity from which we have to extract something is the denominator. For example, when we say 4/5, we are essentially extracting four parts out of five.
4. Extending the above concept, the quantity which is extracted is our numerator
Example: 15/450 = 1/30
15 is the numerator because we have extracted 15 from 450 and the denominator is 450 because 15 is extracted from 450 so we can say that 1/30^th part of 450 is 15
Tool tip 2: Properties of fractions
Property 1: If we multiply the numerator and denominator by same quantity, the basic value of fraction will never change.  For example: 4/5 x 5/5 = 20/25 = 4/5
Property 2: If there are two fractions a/b and c/d then a/b=c/d when ad=bc. For example 3/4 = 12/16 because 3 x 16 = 4 x 12
Property 3: A fraction with zero as the denominator is not defined.
Property 4: If the numerator of the fraction is zero, then the fraction equals zero.
Property 5: If the numerator and denominator of the fraction are equal, then the fraction is equal to one.
DECIMALS
Are all numbers integers? Well, the obvious answer to that question is a no. All numbers are not integers. Consider the case of 0.333333. What is this number? An integer? Well, it is a decimal. But wait are decimals? Decimals are nothing else but the values lying between two integers on the number line.
Remember the following decimal forms (showcasing how decimals look like on either side of the number line):
Decimals less than -1: -12.12, -9.13, -1.2 (the numbers are arranged in increasing order with -1.2 being the largest among all).
Decimals between -1 and 0: -0.52, -0.40, -0.04
Decimals between 0 to 1: 0.12, 0.14, 0.80
Decimals greater than 1: 2.13, 5.64, 7.83
Relating decimals to fraction: When we solve a fraction of the p/q form, it is not necessarily it would return an integral value.  When we left with a remainder, we ultimately convert it into decimal form. Some examples of Decimals are 4.5, 9.6, 6.78, and 99.98 , these all are decimal numbers 
Tool tip 1: Forming Decimals from Fractions
How we can write 4/5 in decimal form?
Since the number 4 is smaller than the 5, so the decimal value will be less than one. Multiple and divide both numbers by 10. We have: 40/ (10×5) Which effectively is (dividing the 40 by 5 first)
8/10. Thus, the final result is 0.8 So 4/5 = 0.8. 0.8 is the decimal form.
Tool tip 2: Adding Decimals While adding decimals, you should always write the decimals in a vertical column with the decimal points aligned vertically.
Add all these 0.567 +78+8.9+5.06+56
= 78.000 +  56.00 +   5.06   +  0.567 =  139.627
Tool tip 3: Subtracting Decimal
In addition we can write the numbers in any order. But while subtracting, we should preferably write the numbers in descending order and the vertical column with decimal points should be aligned to the same decimal points.
Let’s take an example: we have to subtract 5.06 from 0.567. We write it as: 5.06 - 0.567 = 5.507
This result is wrong because in case of subtraction we need equal digits in both the quantities, so these blank spaces filled with 0. So this can be done like as 5.060 - 0.567 =  4.493
This is the right approach for the question
Tool tip 4: Multiplying Decimals
As a first step, multiply the given integers in the normal form keeping the decimals aside.  The number of decimal places in the product is then equal to the total of the decimal places in the two decimals. It is as simple as that.
Consider the following example : 5.060 X 0.567  = 2.869020
First multiply 5060 with 567 and get the result as 2869020 and then move the decimal point to 6 places from the left, that is between 2 and 8. Effectively, we move it to six decimals places, our sum total of the decimal places in the two numbers.

cyclicity : 
 type 1 : where power of single digit is considered
The concept of cyclicity is used to identify the last digit of the number which is in the form of power like p^k. Let’s take an example to understand this:
Example 1 :  find the unit digit of 3⁵⁶.
Solution :  Now it’s a big term so we cannot find the last digit by doing 3 x 3 x 3 x 3 x 3……. 56 times so we use the concept of cyclicity
Step 1 : 3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
So now pay attention to the last digits we saw that the last digit repeats itself after a cycle of 4 and the cycle is 3 ,9,7,1,this is called the cyclicity of any number ,therefore when we need to find the unit digit of any number like 3ⁿ we just need to find the number on which the cycle of last digit ends .  And in the next step we will divide the power with the cyclicity

• if the remainder will be 1 then the unit digit will be 3
• if the remainder will be 2 then the unit digit will be 9
• if the remainder will be 3 then the unit digit will be 7
• if the remainder will be 0 then the unit digit will be 1

This is all about the cyclicity
Why the power is divided by number 4.
We will divide the power with 4 because cycle repeat itself after 4 values, and also we need to find the remainder which tells us the required values to complete the next cycle.
Now the main question was that how much is the last digit of 3⁵⁴
So we know the cycle repeats itself after 4 so we will divide the 54 with 4 ,so on dividing 54 by 4 the remainder becomes 2 .Now as we discussed above if the remainder is 2 the last digit would be 9, so in the end the unit digit of 3⁵⁴ is 9.
Type 2 :  where power of 2 and 3 digits number is to be considered
Example 2 : What will be the unit digit of 24⁴⁵ or 347⁴⁵
Solution : Lets take some example to understand it very clearly
We know that unit digit of 3 x 3 = 9
And the unit digit of 453 x 543 = 9
The main purpose of the above expression is that the unit digit of any multiplication depends upon the unit digit of numbers , whatever is the number big or small the unit digit always depends upon the multiplication of the last digit .
So the last digit of 24⁴⁵ can be found by 4⁴⁵
So the cyclicity of 4 is 2 because the cycle of last digit repeats after two values
4¹ = 4
4² =16
4³ = 64
So when we divide 45 with 2 then we will get the remainder as 1 and the last digit will be 4
Now come to the case number second unit digit of 347⁴⁵
The unit digit of this number can be find by the same method
The cyclicity of 7 is  4
7¹  = 7
7² = 49
7³ = 343
7⁴ = 2401
So on dividing 45 with 4 , 1 will be the remainder and the last digit would be 7
Type 3 : where _p^qr is to be considered
What will be the last digit of ₁₂²³⁴⁵
To find the last digit of this type of number we  will start the question from the base the base is 12. It means we will see the cyclicity of 2 because the last digit is depends upon the unit digit of 12. Lets do it step vise step
Before the steps we will write the last digits of
2¹ = 2
2² = 4
2³ = 8
2⁴ = 6
2⁵ = 2
Step 1: Now we know that cyclicity of last digit of 12 i.e 2 is 4 , hence the divide the power of 12 i.e 23⁴⁵  with 4
Step 2:  Now the remainder 23⁴⁵ /4 will determine the last digit.
Step 3:  The remainder will be 3 because we can write remainder of 23 /4 = 3 or -1 and -1 ⁴⁵ / 4 will give us remainder as -1 or 3
Hence in the end the last digit of ₁₂²³⁴⁵ is nothing but 12³  =  8.
How to find unit digit of a number :
For the concept of identifying the unit digit, we have to first familiarize with the concept of cyclicity. Cyclicity of any number is about the last digit and how they appear in a certain defined manner. Let’s take an example to clear this thing:
The cyclicity chart of 2 is:
2¹ =2
2² =4
2³ =8
2⁴=16
2⁵=32
Have a close look at the above. You would see that as 2 is multiplied every-time with its own self, the last digit changes. On the 4^th multiplication, 2⁵ has the same unit digit as 2¹. This shows us the cyclicity of 2 is 4, that is after every fourth multiplication, the unit digit will be two.
Cyclicity table:
The cyclicity table for numbers is given as below:
Number     Cyclicity
1                   1
2                  4
3                  4
4                  2
5                  1
6                  1
7                  4
8                  4
9                  2
10                1

How did we figure out the above?
Multiply and see for yourself. It’s good practice. Now let us use the concept of cyclicity to calculate the Unit digit of a number.
What is the unit digit of the expression  4⁴⁵?
Now we have two methods to solve this but we choose the best way to solve it i.e. through cyclicity
We know the cyclicity of 4 is 2
Have a look:
4¹ =4
4² =16
4³ =64
Here the 4 comes again to the end when the 4 raised to the power of 3 so it is clear that the cyclicity of 4 is 2.  Now with the cyclicity number i.e. with 2 divide the given power i.e. 45/2 what will be the remainder the remainder will be 1 so the when remainder was 1 what was the answer when 4 raised to the power one see first step , yes , 4
So the unit digit in this case is 4.
For checking whether you have learned the topic, think of any number like this, calculate its unit digit and then check it with the help of a calculator.
Lets solve another example:
The digit in the unit place of the number 7⁹⁵ X 3⁵⁸ is
A.  7
B.  2
C.  6
D.  4
Solution
The Cyclicity table for 7 is as follows:
7¹ =7
7² =49
7³ = 343
7⁴ = 2401
Let’s divide 95 by 4: the remainder is 3.
Thus, the last digit of 7⁹⁵ is equals to the last digit of 7³ i.e. 3.
The Cyclicity table for 3 is as follows:
3¹ =3
3² =9
3³ = 27
3⁴ = 81
3⁵ = 243
Let’s divide 58 by 4, the remainder is 2. Hence the last digit will be 9.