There are two main approaches to thinking about a number base. The first is what we may call the "Counting to ten" approach.
In a number base, the radix is the base value (e.g. ten, for base-10). A counting system has as many values for each digit as its radix, including 0. For radixes
higher than 9+1, letters are used as additional numerals. Thus, the numerals in hexadecimal [base-16] are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.
Using the counting approach to number bases, one counts up, starting at 0. When you reach the highest numeral included in the counting system,
the next number resets the digit to zero and adds one to the digit immediately to its left. You can see a visual demonstration of this in action in the applet
at mathisfun.com
The second approach to thinking about counting bases is less intuitive but more useful. Consider the number 532. This number is the same
as 500+30+2, and we can also represent this sum as \((5*100)+(3*10)+(2*1)\). This is why we sometimes refer to the positions of digits in a number as the ones place,
tens place, hundreds place, etc. One further generalization of this sum is possible: \[532=(5*10^2 )+(3*10^1 )+2*10^0\] (remember, anything to the power of zero equals one.)
From this exercise we can derive another interpretation of numbers in positional notation. In a base-n system, each digit is equal to
its value multiplied by a successive power of its radix. So 10010 in binary [base 2] is \((1*2^4 )+(0*2^3 )+(0*2^2 )+(1*2^1 )+(0*2^0 )=16+0+0+2+0=18\) in base-10.
We'll use this visualization method for converting numbers between bases below.
Two Plus Two is Ten: Converting between Bases
Converting numbers to and from base-10 are relatively simple operations. Converting between two non-decimal bases
[that is, two bases that are not base-10] is so much more complicated that the preferred method for doing so is generally simply to convert to base-10 and then convert
again to the new base. We will cover the two simple operations below:
If one wants to convert a number from base-10 to base-n, the first step is to divide your number by the radix of the system you are converting
to. This will result in a quotient and a remainder. Write down the remainder. Now repeat the process, dividing your quotient by the radix and writing down the new remainder.
Continue this process until your quotient is zero. Finally, write your remainders in reverse order. This string of digits is your original number in your new base.
So, for example, if we want to convert 232 into base-7, we begin by dividing \(232/7=33\), with a remainder of 1. Now, we repeat the process using
33: \(33/7=4\), with a remainder of 5. In the third iteration, \(4/7=0\) with a remainder of 4. Therefore, 232 in base-7 is 451.
To convert a number to base 10, we use the process from the previous section: write the number as a sum of powers of the radix, and then
simply evaluate this sum in base 10.
using our same example from before, but in reverse, we write 451 in base seven as: \[(1*7^0)+(5*7^1)+(4*7^2)=1+35+196=232\]
You can try this out for yourself, and check your answers using a conversion calculator here .
Right of the Decimal Point: Fractions and Decimals
We've covered how number base systems work with integers (whole numbers), but what about fractions? At this point, the counting visualization fails
us, but the exponent method still holds up. Recall that \(x^{-2}=1/x^2\). So in base-10, where \(0.3=3/10\), we can expand this into \(3/10=3*1/10=3*10^{-1}\).
Likewise, \(0.03=3/100=3*1/10^2 =3*10^{-2}\). Each digit further right of the decimal point decreases the exponent on the radix by 1.
Of course, in non-decimal systems this can become quite complex, and oftentimes very simple decimals in base-10 end up being irrational in other
bases. Of course, the reverse is also true: there are cases where a number which is irrational in base-10, such as 1/3, become rational numbers in other
bases (1/3=0.1 in base-3). This has the potential to make some calculations easier, which we discuss further here .
One Thousand Fingers: Counting in Binary
One interesting, non-digital way to work with binary involves representing binary 1s and 0s using fingers extended or curled. This system of
counting, which is eloquently explained by Squirrel Girl in the comic below, allows the user to count to 1,031 using only the fingers of their hands [click images to enlarge].
Since base-10 owes much of its ubiquitousness to being the amount one can count on their fingers, it's conceivable that this method could be used
to launch a base-1031 number system, and so on. However, this would likely require the use of a nested or mixed-base system, which is discussed, along with other uses of
positional notations, here .
