In ordinary arithmetic we use ten digits or one-symbol numbers {0,1,2,3,4,5,6,7,  8,9} to write all the possible numbers. The symbol for “ten” is 10, meaning “once ten plus zero times one.” For example, 243 means “twice ten-squared plus four times ten plus three.” In symbols, we could write

                                    243 = 2×100+4×10+3= 2×10² +4×10¹+3×10⁰.

To write numbers less than 1, we write 1/10, or 10⁻¹, as .1; 1/100 = 10⁻² = .01,  and so on. Ten is called the base.

In general, suppose a₀,a₁,a₂ and b₁,b₂,b₃ are any digits. When we write the number ...a₂a₁a₀.b₁b₂b₃ ... it means

                     ...+a₂ ×10² +a₁x10¹ +a₀ ×10⁰ +b₁ ×10⁻¹ +b₂ ×10⁻² +b₃ ×10⁻³ +...

Most people believe that we use 10 as the base of our number notation because people have 10 fingers and thumbs on their hands. But there is no special mathematical reason for choosing base 10. Historically, base 60 was used first, by the Sumerians and Babylonians.

We shall consider one other base, the base 2, because it arises in computer applications. Numbers written in base 2 are called binary numbers. To write binary numbers, only the two digits 0 and 1 are necessary.

We shall denote the base by putting the number in parentheses and then putting the base as a subscript. In that notation, (101.11)₂ means 1×2² +0×2¹+1×2⁰+1×2⁻¹+1×2⁻², or in regular (base 10) notation 4+0+1+.5+.25, equaling 5.75.

So we could say (101.11)₂ = (5.75)₁₀. But we will usually omit the parentheses and subscript when the numbers are written in base 10.

Sample Problem 1.1 What is (10111)₂ in base 10?

Solution. (10111)₂ = 1×2⁴ +0×2³+1×2²+1×2¹+1

                                = 16+0+4+2+1

                                 = 23

Sample Problem 1.2 What is (.101)₂ in base 10? What is (10111.101)₂ in base 10?

Solution. (.101)₂ = 1×2⁻¹+0×2⁻²+1×2⁻³

                                = 1×.5+0×.25+1×.125

                            = .5+.125 = .625

Using this and the previous sample problem,

(10111.101)₂ = (10111)₂ + (.101)₂

                        = 123+.625 = 23.625

In order to convert from base 10 to base 2, use continued division until you reach quotient 1, and record the remainders. Start with the final quotient (the 1) and read the remainders upward.

Sample Problem 1.3 What is 108 in base 2?

Solution. 108/2 = 54, remainder 0

                      54/2 = 27, remainder 0

                     27/2 = 13, remainder 1

                       13/2 = 6, remainder 1

                            6/2 = 3, remainder 0

                 3/2 = 1, remainder 1

So you follow the initial 1 with 1101100, and 108 = (1101100)₂.

Conversion of non-integers from base 10 to base 2 can also be done, but is more difficult.