Part One: Hardware world
Chapter 1: Data representation on computer hardware

There are 3 methods used to convert decimal numbers to any base:

1. Successive division method
2. Subtraction method

In this lesson, you will learn how to convert decimal numbers to any base number system using the division method.

We are still in chapter one of CS50AP.

## 1.2.2.2) Conversion from the decimal number system to any base:

### a) Decimal to binary:

#### 1) Successive divisions method:

This is the formal method found in any reference and the one that you will use to solve problems in your exams.

Let’s take an example from which we will extract the general algorithm.

Example 1:

We will convert the integer part and fractional part independently.

##### Conversion of the integer part:

During the conversion from binary to decimal, we were multiplying each binary digit by 2 raised to the power of the digit’s index. So, during the conversion from decimal to binary, we will do the reverse process. We will divide the decimal number by 2.

Steps to convert the integer part

1. Step 1 Divide the integer part by 2
2. Step 2 – Write the remainder in the remainder column and the integer part in the Integer column. If there is no remainder after the division process, then write 0 in the remainder column.
3. Step3 – Repeat the previous 2 steps for the integer part till we reach 0.

Let’s apply these steps to this example:

1. We divided 21 by 2. Why 2? Because it is the base of the binary number system, which is the number system we convert the decimal number to. The result of dividing 21 by 2 is 10.5. Because the result of the division process contains a fraction, then there is a remainder which equals 1. Then, write 1 in the remainder column and 10 in the integer column.
2. Then, divide 10 by 2. The result will be 5. There is no fractional this time. Then, write 0 in the remainder column and 5 in the integer column.
3. Then, divide 5 by 2. The result is 2.5. This time the result contains a fractional part. Then, write 1 in the remainder column and 2 in the integer column.
4. So, divide 2 by 2. The result is 1 without a fraction. So, the remainder is 0 and the integer is 1.
5. Finally, divide 1 by 2. This time the integer part is 0 and this marks the end of the division process. The fractional is 0.5 which gives us 1 as a remainder. Generally, the remainder of any fraction is the numerator.

This means that we continue the division process of the integer part by the base of the number system we convert to until the integer part becomes 0.

###### How to find the remainder easily:
• It is simple. Just multiply the fractional part resulted from the division process by 2 (i.e., by the base of the number system). For example, in the first division process, the result is 10.5, then, the remainder equals .
• If the numerator is less than the denominator, then, the remainder is numerator and the integer part equal 0.

Notice that because you convert to the binary number system, the remainder always equals 0 or 1.

##### Conversion of the fractional part.

During the conversion from the binary number system to the decimal number system, the fractional part was divided by the 2 (i.e., the base of the binary system). So, during the conversion from the decimal number system to the binary number system, we do the reverse process. We multiply the fractional part by 2.

Steps to convert the fractional part:

1. Step 1 – Multiply the fractional part by 2.
2. Step 2 – Write the integer part in the integer column, and the fractional part in the fraction column.
3. Step3 – Repeat the previous 2 steps for the fractional part till we reach 0, or till we reach the requested precision.

Let’s apply these steps to this example:

1. The fractional is 0.25. So, 0.25 x 2 = 0.5.
2. Now, write the integer part to the integer column in the table. Then, continue the multiplication process of the fractional part.
3. So, 0.5 x 2 =1.00. This time the fractional part is 0 which marks the end of the multiplication process.
4. Write the integer part in the integer column.

Notice that the integer part is always 0 or 1.

How to write the final result from these tables:

First, put the decimal point

______.___

Then, in the integer part table, the remainder column represents the equivalent binary number of the integer part. To write it, start from the top remainder to the bottom one as shown by the blue arrow. Write the first remainder next to the decimal point then goes from right to left.

In the fractional part table, the integer column represents the equivalent binary number for the fractional part. To write it, start from top to bottom, and write the first integer number next to the decimal point and goes from left to right.

Now, we can write

(21.25)10 = (10101.01)2

So, what about conversion from the decimal number to any base system? The steps are the same except that we will divide by the base of the number system we convert to. So, in the following examples, we will not explain all the steps in detail, as in the previous example.

### b) Decimal to ternary:

I will show you one example only, but if you want to see more examples, you can watch the appendix section of the playlist on our channel.

Notice that we use the previous method for the conversion from the decimal number system to any number except that we change the base according to the number system we convert to.

For example, instead of dividing or multiplying by 2, we will change it to the base of the system we need to convert to.

#### 1) The Successive division method:

Example 2:

What is the equivalent ternary number of the decimal number (59.39)10?

Steps to convert the integer part:

1. We will divide the integer part by 3 and write the remainder in the remainder column.
2. Continue the division process until you reach ​$$\frac{2}{3}$$​. The integer part is 0 which marks the end of the division process, and the fractional part is $$\frac{2}{3}$$(0.666666). So, the remainder is 2. (the remainder of a fraction is the numerator)

How to find the remainder easily:

• You can get the remainder by multiplying the fractional part resulting from the division by 3. For example, in the first division, $$\frac{59}{3} = 19 \frac{2}{3}$$. If you multiply the fractional part by 3, then the remainder will be 2.
• If you need to know how to get the fractional part in this way, you can watch the video on our channel to see how to get this using the calculator. Also, In the playlist, search in the appendix section how to use the calculator to do all these conversions.
• What if you do not have an advanced calculator that you cannot get the fractional part in this way? It is simple. For example, ​$$\frac{59}{3}=19.66666$$​ . Just subtract$$19.66666 – 19$$, to remove the integer part, then multiply 0.66666 by 3. You will get 2 also.
• The same in $$\frac{19}{3}=6.3333333$$. So, if you do $$6.33333 – 6$$=0.3333, then multiply 0.3333 by 3, the remainder will be 1.

Steps to convert the fractional part :

1. We will multiply the fractional part by 3 and write the integer part in the integer column.
2. Continue the multiplication process. However, there is something new. In this example, you will never reach zero. You will continue in the multiplication process forever. So, when will you stop?

If you are a student, your professor must determine the precision as you will see in the example.

Now, the question is, when will the computer stop? Of course, the computer cannot continue forever, because these needs a computer with a memory increases forever. This is impossible of course.

But each computer is designed to store a certain number of bits, and it stops at this number of bits. If you do not understand this now, do not worry, we will talk about it in the next lessons

Now, we can write

(59.39)10 = (2012.10111)3

In the integer part: the remainder is always from 0 to 2.

In the fractional part: the integer part resulting from the multiplication is always from 0 to 2.

### c) Decimal to Octal:

Here, we will not explain the example in detail as before, but you can watch the video on the channel.

#### 1) The Successive division method:

Example 3:

What is the equivalent octal number of the decimal number (59.39)10?

(59.39)10 = (73.30753)8

In the integer part: the remainder is always from 0 to 7.

In the fractional part: the integer part resulting from the multiplication is always from 0 to 7.

Here, we will not explain the example in detail as before, but you can watch the video on the channel.

#### 1) The Successive division method:

Example 3:

What is the equivalent hexadecimal number of the decimal number (59.39)10?

(59.39)10 = (3B.63D70)16

Notice that this time the remainder is 11 because we convert to the hexadecimal number system. So, 11 is not written as 11 but it is written B. As we said before, we cannot consider 11 as one digit, so, from 10 we give letters as follow

A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

The same in the fractional part, 13 is written as D.

In the integer part: the remainder is always from 0 to 15.

In the fractional part: the integer part resulting from the multiplication is always from 0 to 15.

#### The general algorithm for converting decimal number system to any base number system using the division method:

Because we used the same method in all previous examples, we can extract a general algorithm for conversion from the decimal number system to any other number system.

##### Converting the integer part:
1. Divide the integer part by the base (n) of the number system we convert to.
2. Calculate the remainder in each division process by multiplying the resulting fractional part by the base (n) of the number system we convert to. Then write the remainder in the remainder column.
3. Continue in the division process until the resulting integer part becomes less than the base (n).
4. Write the remainder from top to bottom and from right to left starting next to the decimal point.
##### Conversion of the fractional part:
1. Multiply the fractional part by the base (n) of the number system we convert to.
2. Write the resulting integer part in the integer column.
3. Continue the multiplication process of the fractional part and repeat step 2 each time.
4. Write the integer part from top to down and from left to right starting from the decimal point.

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