NUMERAL SYSTEMS A decimal system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by civilizations. For example the decimal 5678 represents 5 × 103 + 6 × 102 + 7 × 101 + 8 × 100 = 5 x 1000 + 6 x 100 + 7 x 10 + 8 It generally uses a new symbol for each new power of 10. Then 1 10 100 1000 have a different symbol. For instance, Ancient Egyptians used these numerals, written in hyeroglyphic script. They used two ones to represent two, three ones for three, and so on, up to nine. Write 5678 using Ancient Egyptians numerals. Problems : - It may take a long time to write a number. - You need a new symbol for any new power of 10. As another example, Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). I 1 Write 5678 using Roman numerals. V 5 X 10 L 50 C 100 D 500 Problem : - It takes less time to write a number but you still need a new M 1000 symbol for any new power of 10. Why 10 ? Probably because ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). But some cultures do, or did, use other bases of numbers. Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (using all twenty fingers and toes). The Yuki language in California and the Pamean languages in Mexico have octal (base-8) systems (because the speakers count using the spaces between their fingers rather than the fingers themselves). The Babylonians used a base-60 system. Others used base-4 system, quinary (base-5) system, base-12 system, base-15 system, base-24 system or base-32 system. A positional system only needs a few symbols to represent any number, no matter how large it is. A positional number system is one where the numbers are arranged in columns and then the value of one symbol depends on its place. Babylonians used stylus made of reed to write cuneiform symbols onto a wet clay tablet, which later they baked if they wished to preserve what was written on it. The number of elementary symbols is very small - there are only few lines put together in different ways - but their combinations give extremely large number of possibilities. Moreover, they tended to arrange the symbols into neat piles. The Babylonian sexagesimal positional system (base-60) places numbers with the same convention as ours, so the right most position is for the units up to 59, the position one to the left is for 60 × n where 1 ≤ n ≤ 59, etc. If we adopt a notation where we separate the numerals by commas so, for example, 1,57,46,40 represents the sexagesimal number 1 × 603 + 57 × 602 + 46 × 60 + 40 which, in decimal notation is 424000. There is a problem : the symbol for sixty seems to be exactly the same as that for one. Sixty one is sixty and one, which therefore looks like one and one, and so on. Surely this is very confusing! x 3600 x 60 Units Value 1 1 + 1 = 2 60 60 + 1 = 61 10 x 60 = 600 3600 (60 x 60) 2 x 3600 = 7200 You can now see why they piled the units up into neat piles! They needed to distinguish one plus one or two, from one times sixty plus one meaning sixty one. Both these have two symbols for one. But the representation of two has the two ones touching, while the representation for sixty one has a gap between them. A careless clerk might make mistakes that way, but if you were careful, it should be all right. Believe it or not, this didn't worry them. After all, if you were counting things, you would tend to know if you were counting individual things or counting in lots of sixty (or even 3,600!) So the Babylonians didn't bother with a zero at the end of the number. However, it is more serious with gaps in the middle of the number. The number 3601 is not too different from 3660, and they are both written as two ones. You could say that there should be a bigger gap for 3601, since the gap represents nothing in the sixty column, but how easy to make a mistake! So the Babylonians DID have a zero, which they used only in only in the middle of numbers. = 60 x 60 = 3,600 = 3,600 + 60 = 3,660 3,600 + 0 + 1 = 3,601 The great advantage of the positional system is that you need only a limited number of symbols (the Babylonians only had two, plus their symbol for zero) and you can represent any whole number, however big. You can also do arithmetic far easier. The Babylonians had a sophisticated number system. Write 5678 using Babylonians numerals. Base 60 in modern times You many wonder why they seemed to like the number sixty so much. Sixty is a very good number for a base. The number of divisors was maximised. 10 is divisible by 2 5 60 is divisible by 2 3 4 5 6 10 12 15 20 30 In fact, the Babylonians have given their base 60 to us. There are 60 minutes in an hour, and 60 seconds in a minute. There are also 360 degrees in a circle (6 x 60), and a single degree can be broken down still further. There are 60 minutes in a degree, and 60 seconds in one of these minutes. (There is no connection between angle minutes & seconds and time minutes & seconds.) Babylonians were the first people to record the calendar, with the day divided into 24 hours (as we do), each hour into 60 minutes and each minute into 60 seconds. We use a positional decimal systems : the Hindu-Arabic numeral system. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. The Hindus in India invented this system from about the 4th century BC onwards. This number system spread to the Middle East in about the 9th century AD, where it was used by Arab mathematicians and astronomers. (Muslim scientists used the Babylonian number system, and merchants used the Abjad numerals, a system based on letters, rather like Greek numbers.) Arabic numbers then spread to Europe. Before this, Europeans were using Roman numbers, with abacuses for calculation. Fibonacci wrote a book about Arabic numbers in the thirteenth century AD. At first, these numbers were very unpopular in Europe, since people were used to using abacuses where you could watch the calculation taking place. But they soon realised how much easier it was to do calculations with Arabic numbers. Now Arabic numbers are generally used throughout the world for calculation. And today computers use a base-2 system also called binary system. With this system, calculations are very easy to compute. Binary addition works exactly like decimal. You line up the numbers, and add up each column, starting from the right. However, it is much easier, as the only possible additions are 0+0=0, 0+1=1, 1+0=1, 1+1=10. You do need to carry, but that doesn't complicate things much. Binary multiplication doesn't involve any multiplication tables at all. All you need are 0x0=0, 1x0=0, 0x1=0, 1x1=1. For long multiplication, you do need to add. It works like decimal multiplication. You separate one number into its powers of two (if it was a decimal number, this would be separating the number into its hundreds, tens and units, etc., so 3426 = 3000 + 400 + 20 + 6). Then you multiply the other number by each digit. You need the add the right number of '0's on the end, of course. .