Math Problem Solver

Numeration Systems

Throughout history each culture has developed its own way to keep track of numbers. Each of

these numeration systems has some distinct features, and some features in common with other

systems. First, each system uses special words or symbols to represent numbers. However, in

deciding what words or symbols you want to use, you also have to decide whether location or order

of the words/symbols affects meaning. Most numeration systems can be classified as one of the

following two types of system:

Symbol value: each symbol represents the same number, regardless of where it appears.

example: Roman numerals — the Vs in XVIII and MCMLV always mean five

Place value: each symbol represents groups of a size that depends on where it appears.

example: Hindu-Arabic numerals — the 5s in 135 and 5042 mean different amounts

Historical note: The original Roman system used IIII for four and VIIII for nine. The inversion

that led to IV and IX was a later innovation which we shall not use here.

1. Convert the following base ten numerals into Roman numerals, and then find their sum

without converting back to base ten: 64, 137, 300.

2. The Babylonians used a sexagesimal (base-sixty) positional numeral system inherited from

the Sumerians (whom they replaced about 2000 BC). Babylonian numerals were written in

cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would

be exposed in the sun to harden to create a permanent record (or recycled before hardening).

The Sumerian-Babylonian system used only two symbols: a vertical (downward) wedge to

represent the number one, and a horizontal (leftward) wedge to represent ten. Sexagesimals

still survive to this day, in the form of degrees, minutes, and seconds in the measurement of

time and angles.

Here are the Babylonian numerals for fifty-eight, fifty-nine, sixty, sixty-one, and sixty-two:

Explain how to interpret each of these numerals. Note how the ones and tens are stacked in

the first two numerals. How is place value invoked in the latter three numerals?

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3. Convert the following base ten numerals into Babylonian numerals, and then find their sum

without converting back to base ten: 64, 137, 300.

4. List the advantages and disadvantages of representing and adding numbers in each of the two

types of system.

5. The ancient Babylonian tablet below gives the multiplication tables for what number?

(The symbols in the first two columns [except for the first row] mean “the number multiplied

by”.)

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