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 \(16\text{.}\)
In the video in Figure 11.19 introduce numbers represented in bases other than \(2\) and \(10\text{.}\) Further details are give below.
Figure11.19.Base \(b\) Numbers by Matt Farmer and Stephen Steward
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 \(99\text{.}\)
English
French
binary (base 2)
ternary (base 3)
octal (base 8)
decimal (base 10)
dozenal (base 12)
hexa-decimal (base 16)
zero
zéro
\(\mathrm{0}_2\)
\(\mathrm{0}_3\)
\(\mathrm{0}_8\)
\(\mathrm{0}\)
\(\mathrm{0}_{12}\)
\(\mathrm{0}_{16}\)
one
un
\(\mathrm{1}_2\)
\(\mathrm{1}_3\)
\(\mathrm{1}_8\)
\(\mathrm{1}\)
\(\mathrm{1}_{12}\)
\(\mathrm{1}_{16}\)
two
deux
\(\mathrm{10}_2\)
\(\mathrm{2}_3\)
\(\mathrm{2}_8\)
\(\mathrm{2}\)
\(\mathrm{2}_{12}\)
\(\mathrm{2}_{16}\)
three
trois
\(\mathrm{11}_2\)
\(\mathrm{10}_3\)
\(\mathrm{3}_8\)
\(\mathrm{3}\)
\(\mathrm{3}_{12}\)
\(\mathrm{3}_{16}\)
four
quatre
\(\mathrm{100}_2\)
\(\mathrm{11}_3\)
\(\mathrm{4}_8\)
\(\mathrm{4}\)
\(\mathrm{4}_{12}\)
\(\mathrm{4}_{16}\)
five
cinq
\(\mathrm{101}_2\)
\(\mathrm{12}_3\)
\(\mathrm{5}_8\)
\(\mathrm{5}\)
\(\mathrm{5}_{12}\)
\(\mathrm{5}_{16}\)
six
six
\(\mathrm{110}_2\)
\(\mathrm{20}_3\)
\(\mathrm{6}_8\)
\(\mathrm{6}\)
\(\mathrm{6}_{12}\)
\(\mathrm{6}_{16}\)
seven
sept
\(\mathrm{111}_2\)
\(\mathrm{21}_3\)
\(\mathrm{7}_8\)
\(\mathrm{7}\)
\(\mathrm{7}_{12}\)
\(\mathrm{7}_{16}\)
eight
huit
\(\mathrm{1000}_2\)
\(\mathrm{22}_3\)
\(\mathrm{10}_8\)
\(\mathrm{8}\)
\(\mathrm{8}_{12}\)
\(\mathrm{8}_{16}\)
nine
neuf
\(\mathrm{1001}_2\)
\(\mathrm{100}_3\)
\(\mathrm{11}_8\)
\(\mathrm{9}\)
\(\mathrm{9}_{12}\)
\(\mathrm{9}_{16}\)
ten
dix
\(\mathrm{1010}_2\)
\(\mathrm{101}_3\)
\(\mathrm{12}_8\)
\(\mathrm{10}\)
\(\mathrm{A}_{12}\)
\(\mathrm{A}_{16}\)
eleven
onze
\(\mathrm{1011}_2\)
\(\mathrm{102}_3\)
\(\mathrm{13}_8\)
\(\mathrm{11}\)
\(\mathrm{B}_{12}\)
\(\mathrm{B}_{16}\)
twelve
douze
\(\mathrm{1100}_2\)
\(\mathrm{110}_3\)
\(\mathrm{14}_8\)
\(\mathrm{12}\)
\(\mathrm{10}_{12}\)
\(\mathrm{C}_{16}\)
thirteen
treize
\(\mathrm{1101}_2\)
\(\mathrm{111}_3\)
\(\mathrm{15}_8\)
\(\mathrm{13}\)
\(\mathrm{11}_{12}\)
\(\mathrm{D}_{16}\)
fourteen
quatorze
\(\mathrm{1110}_2\)
\(\mathrm{112}_3\)
\(\mathrm{16}_8\)
\(\mathrm{14}\)
\(\mathrm{12}_{12}\)
\(\mathrm{E}_{16}\)
fifteen
quinze
\(\mathrm{1111}_2\)
\(\mathrm{120}_3\)
\(\mathrm{17}_8\)
\(\mathrm{15}\)
\(\mathrm{13}_{12}\)
\(\mathrm{F}_{16}\)
sixteen
seize
\(\mathrm{10000}_2\)
\(\mathrm{121}_3\)
\(\mathrm{20}_8\)
\(\mathrm{16}\)
\(\mathrm{14}_{12}\)
\(\mathrm{10}_{16}\)
seventeen
dixsept
\(\mathrm{10001}_2\)
\(\mathrm{122}_3\)
\(\mathrm{21}_8\)
\(\mathrm{17}\)
\(\mathrm{15}_{12}\)
\(\mathrm{11}_{16}\)
twenty
vingt
\(\mathrm{10100}_2\)
\(\mathrm{202}_3\)
\(\mathrm{24}_8\)
\(\mathrm{20}\)
\(\mathrm{18}_{12}\)
\(\mathrm{14}_{16}\)
sixty
soixante
\(\mathrm{111100}_2\)
\(\mathrm{2020}_3\)
\(\mathrm{74}_8\)
\(\mathrm{60}\)
\(\mathrm{50}_{12}\)
\(\mathrm{3C}_{16}\)
eighty
quatrevingt
\(\mathrm{1010000}_2\)
\(\mathrm{2222}_3\)
\(\mathrm{120}_8\)
\(\mathrm{80}\)
\(\mathrm{68}_{12}\)
\(\mathrm{50}_{16}\)
ninety
quatrevingt-dix
\(\mathrm{1011010}_2\)
\(\mathrm{10100}_3\)
\(\mathrm{132}_8\)
\(\mathrm{90}\)
\(\mathrm{76}_{12}\)
\(\mathrm{5A}_{16}\)
hundred
cent
\(\mathrm{1100100}_2\)
\(\mathrm{10201}_3\)
\(\mathrm{144}_8\)
\(\mathrm{100}\)
\(\mathrm{84}_{12}\)
\(\mathrm{64}_{16}\)
Figure11.20.Selected numbers in English, French, and bases \(2\text{,}\)\(3\text{,}\)\(8\text{,}\)\(10\text{,}\)\(12\text{,}\)\(16\)
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
where \(n\) is the number of digits in the base \(b\) representation of \(a\) and \(0\le r_i\lt b\) for \(i\in\{0,\dots,n-1\}\text{.}\)
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 \(10\text{.}\)
\(a\) in
base \(7\) digits of \(a\)
base \(7\) expansion of \(a\)
\(a\) in
base \(7\)
\(7^3\)
\(7^2\)
\(7^1\)
\(7^0\)
base 10
\(1_7\)
\(1\)
\(1\cdot 1\)
\(1\)
\(10_7\)
\(1\)
\(0\)
\(1\cdot 7+0\cdot 1\)
\(7\)
\(100_7\)
\(1\)
\(0\)
\(0\)
\(1\cdot 7^2+0\cdot 7+0\cdot 1\)
\(49\)
\(200_7\)
\(2\)
\(0\)
\(0\)
\(2\cdot 7^2+0\cdot 7+0\cdot 1\)
\(98\)
\(6200_7\)
\(6\)
\(2\)
\(0\)
\(0\)
\(6\cdot 7^3+2\cdot 7^2+0\cdot 7+0\cdot 1\)
\(341\)
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 \(6\text{.}\)
\(a\) in
base \(16\) digits of \(a\)
base \(16\) expansion of \(a\)
\(a\) in
base \(16\)
\(16^3\)
\(16^2\)
\(16^1\)
\(16^0\)
base 10
\(1_{16}\)
\(1\)
\(1\cdot 1\)
\(1\)
\(\mathrm{C}_{16}\)
\(\mathrm{C}\)
\(12\cdot 1\)
\(12\)
\(10_{16}\)
\(1\)
\(0\)
\(1\cdot 16+0\cdot 1\)
\(16\)
\(\mathrm{A}0_{16}\)
\(\mathrm{A}\)
\(0\)
\(10\cdot 16+0\cdot 1\)
\(160\)
\(\mathrm{FF}_{16}\)
\(\mathrm{F}\)
\(\mathrm{F}\)
\(15\cdot 16+15\cdot 1\)
\(255\)
\(100_{16}\)
\(1\)
\(0\)
\(0\)
\(1\cdot 16^2+0\cdot 16+0\cdot 1\)
\(256\)
\(200_{16}\)
\(2\)
\(0\)
\(0\)
\(2\cdot 16^2+0\cdot 16+0\cdot 1\)
\(512\)
\(6\mathrm{B}00_{16}\)
\(6\)
\(\mathrm{B}\)
\(0\)
\(0\)
\(6\cdot 16^3+11\cdot 16^2+0\cdot 16+0\cdot 1\)
\(27392\)
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 \(\mathrm{F}_{16}=15\text{.}\)
We compute the decimal representation of a base \(b\) number by evaluating its base \(b\) expansion.
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.
Problem11.24.Convert from base \(18\) to base \(10\).
Give the base \(18\) expansion of \(99GD872_{18}\) and covert \(99GD872_{18}\) to decimal representation.
Solution.
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
\(0_{18} = 0\)
\(1_{18} = 1\)
\(2_{18} = 2\)
\(3_{18} = 3\)
\(4_{18} = 4\)
\(5_{18} = 5\)
\(6_{18} = 6\)
\(7_{18} = 7\)
\(8_{18} = 8\)
\(9_{18} = 9\)
\(A_{18} = 10\)
\(B_{18} = 11\)
\(C_{18} = 12\)
\(D_{18} = 13\)
\(E_{18} = 14\)
\(F_{18} = 15\)
\(G_{18} = 16\)
\(H_{18} = 17\)
So as the base 18 expansion of \(99GD872{18}\) we get
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 \(b\)
evaluate this expression to obtain the base \(10\) representation.
Try yourself.
Checkpoint11.25.Other base less than \(10\) to base \(10\).
Give the expanded base \(9\) form of \(548725_{9}\text{.}\) Enter all digits in decimal form, that is, for \(A\) enter \(10\text{.}\)
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.
Checkpoint11.26.Other base greater than \(10\) to base \(10\).
Give the expanded base \(19\) form of \(89BDIG5_{19}\text{.}\) Enter all digits in decimal form, that is, for \(A\) enter \(10\text{.}\)
