Section 4.2 Binary Conversions - Table Method

Almost always, there is more than one possible algorithm to solve a problem; converting numbers from Binary to Decimal and back is no exception. We are going to look at two different approaches: use of a table of powers of 2 and repeated multiplication/division by 2. Although the table method shown on this page is the most straightforward one, the division/multiplication method shown on the next page is important to understand because it is easier to express as an algorithm that a computer can execute.

Subsection 4.2.1 Binary to Decimal

To convert a binary value to a decimal one, start with a table of binary column values. The rightmost digits of the number should be in the “ones” column and each additional column to the left should be twice the value of the previous column: … 64, 32, 16, 8, 4, 2, 1. Write your number in the columns so the last digit is in the “ones” column. The decimal value of your number is the sum of each of the columns that has a 1 bit in it.

For example, I want to convert \({11010}_{2}\) to decimal. I will need at least five columns to hold the five bits. They will be numbered: 1s, 2s, 4s, 8s, 16s. When I write those columns in descending order and write my number it looks like this:

Table 4.2.1. "11010" base 2

16s	8s	4s	2s	1s
1	1	0	1	0

That tells me I need to add 16, 8 and 2. So \({11010}_{2} = 16 + 8 + 2 = {26}_{10}\)

Subsection 4.2.2 Decimal to Binary

The first way to convert a decimal number to a binary one is with a table like the one below (if needed you can add more columns - each new column to the left should be twice the value of the preceding one). Then, working from left to right, decide if you need that column’s value to make your number. The value for any column you use should be subtracted from the value you are trying to make.

Table 4.2.2. "10011011" base 2

128s	64s	32s	16s	8s	4s	2s	1s
?	?	?	?	?	?	?	?

For example, I want to write 155 as a binary number. Here is what my thought process might look like:

Do I need 128? Yes, that is less than 155. 
               155 - 128 = 27: that is what we have left to make
Do I need 64? No - I only have 27 left
Do I need 32? No - I only have 27 left
Do I need 16? Yes, that is less than 27. 
              27 - 16 = 11: that is now what we have left
Do I need 8? Yes, that is less than 11.
             11 - 8 = 3
Do I need 4? No - I only have 3 left
Do I need 2? Yes. 
             3 - 2 = 1 
Do I need 1? Yes. 
             1 - 1 = 0 left - we are done!

If we think of each column we used as representing a 1 and each column we didn’t as a 0, we get this:

Table 4.2.3.

128s	64s	32s	16s	8s	4s	2s	1s
1	0	0	1	1	0	1	1

Or written without a table: \({10011011}_{2}\text{.}\)

Just like with decimal numbers, leading zeros do not modify the value of a number. So \({00000101}_{2}\) is the same as \({101}_{2}\text{.}\)

Checkpoint 4.2.4.

What decimal value does the binary number 111100 represent?

Checkpoint 4.2.5.

How many columns do you need to represent 125?

7
8
The 8th column would be 128… we don’t need that
32
32 bits can represent over 4 billion
128

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