Bases

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Image 1. This picture shows the 0s and 1s that form the symbol library of a base two system.
Image 1. This picture shows the 0s and 1s that form the symbol library of a base two system.

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What is a base-positional system?

A base-positional system is a way of writing numbers that makes written computations easier. In such a system, the meaning of a group of symbols depends on the symbols' positions. For example, the symbol 6 means the same thing in 625 and 699, but it means something different when it appears in 625 than in 2,036. In 625 and 699, 6 represents six hundreds, but in 2,036 it represents six ones.

Contrast this with Roman Numerals, a non-positional system, where the meaning of a symbol is more dependent on what symbols are around it than on the symbol's position. For example, in III, the symbol I has the same value in all three positions. In a positional system, it would have different values in each position. On the other hand, the symbol V means something different in VI than in VL. In VI it means +5, but in VL it means -5, even thought V is in the same position in both numbers.


One of the advantages of positional systems is that they generally allow for simpler algorithms for computing basic arithmetic than other number systems. When Europeans were using Roman Numerals, basic arithmetic was done on abacuses, because there was no commonly known method to quickly do written multiplication or division. Other advantages of positional systems are that they can represent arbitrarily high quantities with a limited set of symbols, and the amount of symbols in a number generally corresponds to the number's magnitude.

Base ten system

The number system that you are probably most familiar with is the so-called Arabic number system (which was actually invented in India). This is the system that you use every day. It's a base ten system, because it has ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). The position a symbol is in determines what power of ten it will be multiplied by.


To count in this system, we start with single digit numbers, counting up 0, 1, 2, 3, and so on. When we reach 9, we run out of symbols, and so we turn to the positional aspect of the system to indicate higher values. The next number, of course, is 10. The 1 placed to the left of the 0 means one set of ten, and not just one as it does in 1.


The number 625 means 6 hundreds, 2 tens, and 5 ones. Each position to the left represents one higher power of ten, the base number. So 625 can also be written as

(6 x 102) + (2 x 101) + (5 x 100)

and 1001 can be written as

(1 x 103) + (0 x 102) + (0 x 101) + (1 x 100).

(Remember that 100 = 1 )


For an example of a base ten number system that is quite different from the Arabic system, see Quipu.

Binary

Binary is probably the next most familiar base-positional system, as it's the system used in computers. As its name suggests, binary is a base two system, which means that it has only two symbols in its library: 0 and 1.


Just like in the base ten system, as we move from right to left, each position represents one higher power of the base number, which is in this case two. So in binary, the number 10 means 1 two and 0 ones, or (1 x 21) + (0 x 20), which is simply two. If we translate the number 1001 from binary to base ten, we get:

(1 x 23) + (0 x 22) + (0 x 21) + (1 x 20) = (1 x 8) + (0 x 4) + (0 x 2) + (1 x 1) = 8 + 1 = 9 .


Binary is used in computing because its two symbols can correspond to physical switches being either on or off.

Other bases

Other cultures with recorded number systems have used different bases.


The Mayans, for example, have elements of both base five and base twenty in their system. Their symbol library consists of a zero-symbol, dots, and lines, which are stacked on top of each other, so that the units place is at the bottom of a column of symbols. Zero is represented by a shell-like shape, the numbers one through four are represented by lines of that many dots, and the number five is represented by a solid horizontal line. Numbers after five are represented by collections of dots on top of lines representing groups of five, which is why we say the system appears to be partially base five. Numbers zero through nineteen are shown in Image 2. At nineteen, the Mayan numbers change again. Because the system is base twenty, twenty is represented by a one on top of a zero, indicating (1 x 201) + (0 x 200) . The number 48 would be written as two dots (to represent two twenties) on top of three dots (representing three ones) on top of a bar (representing five ones). Numbers twenty and forty are shown in Image 3.

Image 2. Mayan numbers zero through nineteen.
Image 2. Mayan numbers zero through nineteen.
Image 3. Mayan numbers twenty and forty eight.
Image 3. Mayan numbers twenty and forty eight.

The Babylonian number system is very much like the Mayan system, but with bases ten and sixty. Numbers one through nine are represented by clumps of their symbol for one (which looks a little like a Y), and at ten there's a new symbol (which looks like a sideways A). The numbers up to fifty-nine are represented by closely packed clumps of tens and ones, as shown in Image 4. At sixty, though, the system starts over - the symbol for sixty is the same as the symbol for one. The number 62 would be written as a one-symbol to the left of two other one-symbols. The symbols that are supposed to be interpreted as two ones are touching each other, and the one that means one sixty is spaced farther apart. The Babylonians had no zero, and so they relied on spacing to indicate when a position was empty, which presumably got confusing sometimes.

Image 4. Babylonian numerals 1 through 60.
Image 4. Babylonian numerals 1 through 60.

Many people believe that the Babylonian system is responsible for the elements of base sixty that we see in our culture, in things like time (60 seconds in a minute, 60 minutes in an hour) and geometry (360 degrees in a circle).

References and further reading

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