R. Craig Collins >Common > How To: Numbering Systems
How To: Numbers © R. Craig Collins, 2005
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
Why are there different numbering systems? Just as different countries use different words for the same item, there are different numbering systems that represent the same thing in a different way.
Why are numbers important? Computer only really use 1s and 0s, so from a theoretical point of view, you should be able to convert our human base 10 numbers to a computer friendly base 2 number. But from a practical point of view: if you are making web pages, and you don't want to be stuck with 16 word colors, you have to be able to express mixtures of red, green, and blue in numbers. The most widely available graphics program is Paint, but it gives the numerical values of red, green, and blue in base 10... but web pages use base 16, hexadecimal. So, just to be able to use a lot of colors, you need to be able to convert the numbers.
So, you should know the process.
But I know what most of you will do, use a calculator to convert; that's fine, IF you understand the theory.
The three rules of modern numbering systems:
Decimal (base 10) 

Binary (Base 2) 
Hexadecimal (Base 16) 
0 

0 
0 
1 

1 
1 
2 

10 
2 
3 

11 
3 
4 

100 
4 
5 

101 
5 
6 

110 
6 
7 

111 
7 
8 

1000 
8 
9 

1001 
9 
10 

1010 
A 
11 

1011 
B 
12 

1100 
C 
13 

1101 
D 
14 

1110 
E 
15 

1111 
F 
16 

10000 
10 
You should be able to recreate the above chart.
There is a video available to demonstrate counting. The link is below, at the end of the building a 'powers of 2 number line' discussion.
Click here for an chart that that goes to 32_{10}, and includes base_{4} and base_{8}.
Notice how large the binary numbers get; we need a way to represent those numbers that takes up less space, and is more distinctive. If you wee to look at the larger example, you can see that base 4, 8, or base 16 are quick ways to represent the same base 2 number with less space. Especially base 16. But decimal, base 10, does not easily fit in this pattern.
Let’s find out why.
Look at the numbers in the binary column. Notice the numbers that begin with one, and are followed by nothing or zeros… and compare those to the decimal equivalent.
1 
10 
100 
1000 
10000 
100000 
1 
2 
4 
8 
16 
32 
The decimal values are doubling. This becomes the basis of a
quick to build
powersoftwo number line.
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
There is a video available on counting, and the Powers of 2 number line here.
This number line is a quick key to converting decimal to binary, and binary to decimal; this is the hardest conversion as the number 10 does not occur on the powers of two number line.
The Harder Stuff, part 1: Converting decimal to binary.
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
115_{10} converts to ?_{2}
The largest number on the powersoftwo number line we can subtract from 115 is 64. Put a one under the 64 to represent "yes, I can subtract this," then do the math… subtract 11564= 51
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 






Can we subtract the next number to the right from 51? Yes! Record the success with a 1, and do the math. 5132 =19
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 
1 





Can we subtract the next number to the right from 19? Yes! Record the success with a 1, and do the math. 1916 =3
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 
1 
1 




Can we subtract the next number to the right from 3? No! Record the failure with a 0, and continue.
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 
1 
1 
0 



Can we subtract the next number to the right from 3? No! Record the failure with a 0, and continue.
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 
1 
1 
0 
0 


Can we subtract the next number to the right from 3? Yes! Record the success with a 1, and do the math. 32 =1
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 
1 
1 
0 
0 
1 

Can we subtract the next number to the right from 1? Yes! Record the success with a 1, and do the math. 11 =0
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 




1 
1 
1 
0 
0 
1 
1 
There are no more subtractions possible, but we have our number! 115_{10} converts to 1110011_{2}
Note: since 115 is odd, it must end in 1.
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
The Harder Stuff, part 2: Converting binary to decimal.
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
11101_{2} converts to ?_{10}
Write the 1s and 0s under the powersoftwo number line.
2^{10} 
2^{9} 
2^{8} 
2^{7} 
2^{6} 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 






1 
1 
1 
0 
1 
Add the value above the successes (1s) and ignore the failures (0s).
16+8+4+1 = 29
We have our number! 11101_{2} converts to 29_{10}
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
The Really Easy Stuff, part 1: Binary to Octal, click here
The Really Easy Stuff, part 2: Binary to Hexadecimal
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
110001011101_{2} converts to ?_{16}
Divide the number into blocks of four, starting from the right. If you don’t have enough numbers, add 0s to the left.
1100 0101 1101
Put a short powersoftwo number line above each of the blocks of four; remember, the 1s and 0s always go under the powersoftwo number line.
8 4 2 1 
8 4 2 1 
8 4 2 1 
1 1 0 0 
0 1 0 1 
1 1 0 1 
12 
5 
13 
12 in hex from the chart , C 
5 in hex from the chart , 5 
13 in hex from the chart , D 
There’s our number! 110001011101_{2} converts to C5D_{16}
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
Hexadecimal, a BAD numbering system, to Binary
BAD_{16}
B 
A 
D 
11 in hex from the chart or 
10 in hex from the chart or 
13 in hex from the chart or 
1011 
1010 
1011 
1011 1010 1101 _{2}
After you have read this, you may wish to click here see how to use a calculator to verify your answers.
_{Just for jollies, Hexadecimal, a BAD numbering system, to Decimal (just use all the skills he have learned)}
_{BAD16}
_{B} 
_{A} 
_{D} 
_{11 in hex from the chart or} 
_{10 in hex from the chart or} 
_{13 in hex from the chart or} 
_{1011 in binary from chart of conversion} 
_{1010 in binary from chart of conversion} 
_{1101 in binary from chart of conversion} 
_{Put 101110101101 under a powersoftwo number line, add the successes to get 298910!}
_{ }_{PRACTICE QUESTIONS}
_{Converting decimal to binary.}
13_{10} converts to ?_{2}
255_{10} converts to ?_{2}
5_{10} converts to ?_{2}
_{Converting binary to decimal.}
1011_{2} converts to ?_{10}
111_{2} converts to ?_{10}
111010_{2} converts to ?_{10}
_{ }_{Converting hexadecimal to binary.}
25_{16} converts to ?_{2}
1F_{16} converts to ?_{2}
F1_{16} converts to ?_{2}
_{Converting hexadecimal to decimal.}
25_{16} converts to ?_{10}
1F_{16} converts to ?_{10}
F1_{16} converts to ?_{10}
You might want to check your answers with the Windows calculator.
See also, a video on Starting the Windows Calculator, or Using the Windows Calculator
First, go to View menu, and choose Scientific View.
In newer versions of Windows, choose Programmer View
Again, in newer versions of Windows, choose Programmer View
If the starting number is in base 10, set the button to
Dec,
enter the number to convert,
then click to bin, oct, or hex.
Example 128_{10}=?_{16}
If the starting number is in base 2, set the button to
bin,
enter the number to convert,
then click to dec, oct, or hex.
If the starting number is in base 16, set the button to
hex,
enter the number to convert,
then click to dec, oct, or bin.
If the starting number is in base 8, set the button to
oct,
enter the number to convert,
then click to dec, hex, or bin.
Exponents
If you are trying to solve something like 2^{8}*2^{8}*2^{8}=2^{24}, and you want to know what 2^{24} is,
Type in 2
then click the x^y button
then type the exponent, 24
then press =, or hit the enter key to calculate