Question

Number Representation and Calculation As you are going through the sections in Chapter 4, you will notice that we are working with different bases. Please explain to the class what it means to: Rewrite a number to a different base Add two or three numbers to a different base Subtract two numbers in a different base Why do we learn to work with numbers in different bases? Please provide examples of your answers. To begin the discussion, click on the "Create Thread" link above.

Answer 1

In mathematics, change of base can mean any of several things:

• Changing numeral bases, such as converting from base 2 (binary) to base 10 (decimal). This is known as base conversion.
• The logarithmic change-of-base formula, one of the logarithmic identities used frequently in algebra and calculus.
• The method for changing between polynomial and normal bases, and similar transformations, for purposes of coding theory and cryptography.

Number in Different Bases

Changing to base 10 from another base

When we write a normal (base 10) number, like 5763, we mean the value:

5000+700+60+3

or, to put it in a more revealing form:

5⋅10³+7⋅10²+6⋅10¹+3⋅10⁰

Notice, the "digits" of our number correspond to the coefficients on the powers of ten that are added together to obtain the value of our number.

In a similar manner, we can specify numbers in other "bases" (besides 10), using different digits that correspond to the coefficients on the powers (of the given base) that must be added together to obtain the value of our number.

For example, the "base 8" (or "octal") number (as indicated by the subscript)

5763₍₈₎

equals

5⋅8³+7⋅8²+6⋅8¹+3⋅8⁰=3059

For smaller bases, we use a subset of these digits. For example, in base 5, we only use digits 0-4; in base 2 (which is also called binary), we only use the digits 0 and 1.

For larger bases, we need to have single digits for values past 9. Hexadecimal (base 16) numbers provide an example of how this can be done. In hexadecimal, we use digits 0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, and F=15. In this way, we have digits corresponding to 0-15, which is what we need.

Changing from base 10 to a different base

One (straight-forward, but inefficient) way to convert from base 10 to a different base is to:

1. Determine the higest power of the base that goes into the number a non-zero number of times.
2. Determine how many times this power can be subtracted from the number without the result being negative (i.e, divide the number by the power). Write this digit down.
3. Redefine the number to be this smallest positive remainder upon division by the power in question
4. Redefine the power to be the power divided by the base.
5. Go back to step 2, unless the power is now less than one -- in which case, you are done.

For example, to convert 1073 to base 5, we recall that:

50 = 1
51 = 5
52 = 25
53 = 125
54 = 625 
55 = 3125

Then we notice that 5⁴ = 625 is the highest power of 5 under 1073.

1073 = 1 * 625 + 448
 448 = 3 * 125 + 73
  73 = 2 * 25 + 23
  23 = 4 * 5 + 3
   3 = 3 * 1 + 0

The bold digits, 13243, reveal the base 5 representation of 1073.

This process, however, is inefficient in that one must both know and use the various powers of the desired base.

There is a simpler way!

Consider the remainders seen upon division of the following numbers by 5:

1073 = 214 * 5 + 3
 214 = 42 * 5 + 4
  42 = 8 * 5 + 2
   8 = 1 * 5 + 3
   1 = 0 * 5 + 1

Note: the base 5 representation comes from reading off the remainders (in red) from bottom to top!
In each step above, we are just dividing by 5 and looking at both the quotient and remainder -- no knowledge of higher powers of 5 is necessary!

Wonderfully, this technique works in any base. (Can you explain why?)

So, for example, if we wanted to find the binary (base 2) representation of 1000, we simply calculate the following:

1000 = 500 * 2 + 0
 500 = 250 * 2 + 0
 250 = 125 * 2 + 0
 125 = 62 * 2 + 1
  62 = 31 * 2 + 0
  31 = 15 * 2 + 1
  15 = 7 * 2 + 1
   7 = 3 * 2 + 1
   3 = 1 * 2 + 1
   1 = 0 * 2 + 1

So 1000 in binary is 1111101000