Instead of using base \(10\) or base \(2\text{,}\) we can use any other natural number \(b>1\) as a base. To represent any number in base \(b\text{,}\) we must specify \(b\) unique symbols that represent the \(b\) values from \(0\) to \(b-1\text{.}\) Those symbols are the first \(b\) symbols from the list
When considering bases \(b\) with \(b\le10\text{,}\) we use the numbers \(0,1,2,3,\dots,b-1\) as our \(b\) unique symbols. However, if \(b>10\text{,}\) we use all of the numbers \(0, 1, 2, \dots, 8, 9\) as well as enough capital letters to complete the list of \(b\) unique symbols. The value of \(\mathrm{A}\) is the decimal number \(10\text{,}\) the value of \(\mathrm{B}\) is the decimal number \(11\text{,}\) the value of \(\mathrm{C}\) is the decimal number \(12\text{,}\) and so on. We do not consider bases greater than \(36\) where we have \(Z_{36}=35\text{,}\) so we do not need further symbols. There are many applications of numbers in other bases. In particular, computer related fields frequently use base \(2\text{,}\)\(8\text{,}\) and \(16\text{.}\)
Figure 11.20 provides various numbers written in base \(2\text{,}\)\(3\text{,}\)\(8\text{,}\)\(10\text{,}\)\(12\text{,}\) and \(16\) as well as in English and French. When counting in some languages, there are irregularities of words which originate in other number systems. In English, the numbers \(11\) and \(12\) do not follow the pattern of the other numbers between \(10\) and \(20\text{.}\) In French, the numbers \(11\) to \(16\) follow a different pattern than the numbers \(17\) to \(19\text{,}\) and the numbers \(30\) to \(79\) follow a different pattern than the numbers \(80\) to \(99\text{.}\)
We generalize the decimal (base 10) expansion to other bases in the following way. Let \(b\in \N\) with \(b>1\text{.}\) We can write any number \(a\in\N\) with \(a\lt b^n\) in the form
To write the number \(a\) in base \(b\text{,}\) we extract the digits \(r_0\) to \(r_{n-1}\) from the expanded notation. To distinguish numbers in different bases, we add a subscript \(b\) to the number in base \(b\) if \(b\ne 10\text{.}\) So, the number \(a\) from above would be written as
in base \(b\text{.}\) In Figure 11.21 and Figure 11.22 we give examples of numbers in base \(7\) and base \(16\) with their digits, expansions, and the numbers in base \(10\text{.}\)
Figure11.21.Numbers in base \(7\text{,}\) their base \(7\) digits, their base \(7\) expansion, and in base \(10\text{.}\) The \(7\) digits used in base \(7\) numbers are \(0\text{,}\)\(1\text{,}\)\(2\text{,}\)\(3\text{,}\)\(4\text{,}\)\(5\text{,}\) and \(6\text{.}\)
Figure11.22.Hexadecimal (base \(16\)) numbers, their base \(16\) digits, their base \(16\) expansion, and in base \(10\text{.}\) The \(16\) symbols used in hexadecimal numbers are \(0\text{,}\)\(1\text{,}\)\(2\text{,}\)\(3\text{,}\)\(4\text{,}\)\(5\text{,}\)\(6\text{,}\)\(7\text{,}\)\(8\text{,}\)\(9\text{,}\)\(\mathrm{A}\text{,}\)\(\mathrm{B}\text{,}\)\(\mathrm{C}\text{,}\)\(\mathrm{D}\text{,}\)\(\mathrm{E}\text{,}\) and \(\mathrm{F}\text{.}\) We have \(\mathrm{A}_{16}=10\text{,}\)\(\mathrm{B}_{16}=11\text{,}\)\(\mathrm{C}_{16}=12\text{,}\)\(\mathrm{E}_{16}=13\text{,}\)\(\mathrm{E}_{16}=14\text{,}\) and \(\mathrm{F}_{16}=15\text{.}\)
Example11.23.Conversion to decimal representation.
Given numbers in various bases \(b\text{,}\) we convert these numbers to their decimal representations by writing out their base \(b\) expansions and then evaluating them.
In base {18} we use the characters \(0\text{,}\)\(1\text{,}\)\(2\text{,}\)\(3\text{,}\)\(4\text{,}\)\(5\text{,}\)\(6\text{,}\)\(7\text{,}\)\(8\text{,}\)\(9\text{,}\)\(A\text{,}\)\(B\text{,}\)\(C\text{,}\)\(D\text{,}\)\(E\text{,}\)\(F\text{,}\)\(G\text{,}\)\(H\) for the digits. The values of these are
So to convert a number in base \(b\) representation, where \(b\) to base \(10\) representation we
write down the base \(b\) expansion, which consists of the digits of the base \(b\) representation converted to decimal and the place values, which are the powers of \(b\)
In Checkpoint 11.26 do the same for a base greater than \(10\text{.}\) Recall that \(\mathrm{A}\) in the base \(b\) representation is a \(10\) in the base \(b\) expansion, \(\mathrm{B}\) in the base \(b\) representation is a \(11\) in the base \(b\) expansion, \(\mathrm{C}\) in the base \(b\) representation is a \(12\) in the base \(b\) expansion, and so on.
