R. Craig Collins >Common > How To: Numbering Systems

How To: Numbers © R. Craig Collins, 2005/22

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:

1. Begin with 0 (zero)
2. The base indicates the number of different characters used to represent the numbers
   (examples base 10 has 10 characters, base two has 2 characters)
3. When you run out of different characters, add a decimal place to the left, and start over.

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 32₁₀, and includes base₄ and base₈.

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.

The decimal values are doubling. This becomes the basis of a quick to build
powers-of-two number line.

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.

115₁₀ converts to ?₂

1. Always build a powers-of-two number line. You don’t need to calculate it, just start with 1, then double!
2. Notice the subscript is getting smaller… that is our clue to ‘subtract’.
3. Computers use 1 to represent success (yes) and 0 to represent failure (no). We will always write our successes and failures (1s and 0s) under the powers-of-two number line.

The largest number on the powers-of-two 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 115-64= 51

Can we subtract the next number to the right from 51? Yes! Record the success with a 1, and do the math. 51-32 =19

Can we subtract the next number to the right from 19? Yes! Record the success with a 1, and do the math. 19-16 =3

Can we subtract the next number to the right from 3? No! Record the failure with a 0, and continue.

Can we subtract the next number to the right from 3? No! Record the failure with a 0, and continue.

Can we subtract the next number to the right from 3? Yes! Record the success with a 1, and do the math. 3-2 =1

Can we subtract the next number to the right from 1? Yes! Record the success with a 1, and do the math. 1-1 =0

There are no more subtractions possible, but we have our number! 115₁₀ converts to 1110011₂

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.

11101₂ converts to ?₁₀

1. Always build a powers-of-two number line. You don’t need to calculate it, just start with 1, then double!
2. Notice the subscript is getting larger… that is our clue to ‘add’.
3. Computers use 1 to represent success (yes) and 0 to represent failure (no). We will always write our successes and failures (1s and 0s) under the powers-of-two number line.

Write the 1s and 0s under the powers-of-two number line.

Add the value above the successes (1s) and ignore the failures (0s).

16+8+4+1 = 29

We have our number! 11101₂ converts to 29₁₀
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.

110001011101₂ 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 powers-of-two number line above each of the blocks of four; remember, the 1s and 0s always go under the powers-of-two number line.

There’s our number! 110001011101₂ converts to C5D₁₆
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

BAD₁₆

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

_{Put 101110101101 under a powers-of-two number line, add the successes to get 298910!}

_{PRACTICE QUESTIONS}

_{Converting decimal to binary.}

13₁₀ converts to ?₂

255₁₀ converts to ?₂

5₁₀ converts to ?₂

_{Converting binary to decimal.}

1011₂ converts to ?₁₀

111₂ converts to ?₁₀

111010₂ converts to ?₁₀

_{Converting hexadecimal to binary.}

25₁₆ converts to ?₂

1F₁₆ converts to ?₂

F1₁₆ converts to ?₂

_{Converting hexadecimal to decimal.}

25₁₆ converts to ?₁₀

1F₁₆ converts to ?₁₀

F1₁₆ 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 128₁₀=?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 2⁸*2⁸*2⁸=2²⁴, and you want to know what 2²⁴ is,
Type in 2
then click the x^y button
then type the exponent, 24
then press =, or hit the enter key to calculate