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1 Ancient ìPositionalî Systems
We have already discussed positional systems. We will now look at two examples of ancient positional systems, the Babylonian system and the Mayan system.
1.1 The Babylonian System
The Babylonian system is a base 60 system, meaning each position represents a power of 60. From what we learned before, this means we need to have symbols that represent digits from 0 to 59. However, the Babylonians did not have a symbol for zero. Therefore, the Babylonians did not have a true positional system. Also, instead of using 59 di§erent symbols to represent the digits 1 to 59, they used combinations of two di§erent symbols.
1.2 Converting from Babylonian to Hindu-Arabic
Since the Babylonians used multiple symbols to represent digits in their system, and since they used no zero symbol, the Babylonian representation of numbers can get quite confusing. To cut down on the confusion, we will put in space between each position. For example, in the numeral below
the space between divides the 60ís position from the 1ís position. In other words, the symbols
are in the 60ís position, and the symbols
are in the 1ís position. Since
represents 10 and
represents 1, the number represented by
is 22 60 + 34 1 = 1354
Example 1 Convert
to Hindu-Arabic.
We see 16 in the 602 ís position, 41 in the 60ís position, and 33 in the 1ís position. So, the value represented is 16 602 + 41 60 + 33 = 60; 093
1.3 Converting from Hindu-Arabic to Babylonian numerals
We convert from base 10 to base 60 in exactly the same way we convert to any other base that is di§erent from 10. We divide the given number by 60 until we cannot divide evenly any more.
Example 2 Convert 1352 to Babylonian numerals.
Dividing successively by 60, we obtain
60 22 R 32
0 R 22
So, we put 22 in the 60ís position and 32 in the 1ís position using Babylonian symbols. So, we get
Example 3 Convert 287,439 to Babylonian numerals.
60 4790 R 39
79 R 50
1 R 19
0 R 1
1.4 Converting from Mayan numerals to Hindu-Arabic
The Mayans worked with a base 20 system...almost. They did have a symbol for zero; however, not every position represents a power of 20. One of the positions is the 360 ís position, or the 18 20 ís position. Since not every position represents a power of the same number, the Mayan system is also not a true positional system. Here are the symbols the Mayans used.
1.5 Converting from Hindu-Arabic to Mayan numerals
In the Mayan numeration system, each position represents a power of 20 (or 18 20 for the special case). So, each time we are going to divide by 20 (with the lone exception of dividing by 18). Letís illustrate this by example.
Example 6 Convert 1327 to Mayan numerals.
We need to Önd out the part of 1327 that 20 does not go into even once. Dividing 1327 by 20 gives
66 R 7
So, the remainder of 7 represents the part of 1327 that 20 did not go into even once. This is the number that goes in the ones position. Now, we need to know what goes into the 20ís position. The next position above the 20ís position is the (18 20)ís position. So, we need to know how many times 18 goes into 66. Remember that 66 represents how many 20ís are in 1327. That is why we are dividing the 66 rather that the 1327 at this stage - we are interested in what goes in the 20ís position.
3 R 12
So, the remainder of 12 represents the part of 1327 that 20 went into once, but 18 did not go into. This is the number that goes in the 20ís position. Finally, we need to know what goes into the (18 20)ís position. The next position above the (18 20)ís position is the
ís position. So, we need to know how many times 20 goes into 3. Well, this is easy, 20 goes into 3 zero times with a remainder of 3. This remainder of 3 represents the part of 1327 that 18 went into and 20 went into once, but that 20 did not go into twice. This is what goes in the (18 20)ís position. Here is the Mayan numeral representation for 1327.
Example 7 Convert 1,650,615 to Mayan numerals.
First, we divide 1,650,615 by 20.
82530 R 15
We could perform the long division, but there is an easier way to get at the remainder by using your calculator. By your calculator, you should see 1650615 20 = 82530: 75 : The 82530 is the quotient, and the 0 : 75 part represents the remainder. If we take 0 : 75 20 ; we get 15, which is the remainder. So, the keystrokes on your calculator you would use to Önd the remainder are
1650615 = 20 ENTER 82530 : 75 82530 ENTER 0 : 75 20 ENTER 15
So, 15 goes in the 1ís position. Next, we divide 82,530 by 18. We get 4585 with a remainder of 0. So, 0 goes in the 20ís position. Next, we divide 4585 by 20. On your calculator, you should get 229 : 25 : Going through the keystrokes described above, you Önd that the remainder is 5. This goes in the (18 20)ís position. Next, we divide 229 by 20. On your calculator, you should get 11 : 45 : This gives a remainder of 9. This goes in the
ís position. Finally, we divide 11 by 20 to get 0 with a remainder of 9. So, 11 goes in the
ís position. Here is the Mayan numeral representation for 1,650,615.
