Origin of Numbers
Today the most commonly used system for writing numbers is the arabic system of numeration: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This system first originated in India and China and was later adopted by arab mathematicians and traders who spread the system further. It wasn't without competition though, since before it there had been a number of other systems. One of these that we still see fairly often today are the Roman Numerals.
Egyptian Numerals
The Egyptian system of numeration is an additive system, which means that the order in which the symbols appear is irrelevant. Their value is always the same and they are simply added together no matter where they are located. This system is one of the oldest and most primitive systems and often quite cumbersome since it often takes a lot of symbols to represent a particular number. This becomes especially true the larger the number gets.
Take this number as an example. It represents the number 389 and in this case the symbols are in the order people who are used to an arabic system would expect, but this is by no means necessarily so.
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Babylonian Numerals
The babylonian numerals were written with a stylus that could create lines and triangles which was also used to write their alphabet: Cuneiform.
The two symbols used are called Nail and Hook respectively and at first glance it looks very much like this system is an additive one as well, and to a certain degree it is. The numbers 1-9 are after all expressed by simply repeating the symbol for 1 the appropriate amount of times. At 10 it then gets the 'hook' symbol to represent ten, which then carries on being doubled for every round of 10. |
At 60 however, something happens and the symbol for 1 reappears. That is because the babylonian system uses 60 as their ”reset point”, the same way we see 100 as a beginning to a new set. Think of how we split up hours into 60 minutes and minutes into 60 seconds. In this area we too operate with 60 being the ”reset point”. To the babylonians this would simply have been applied to all areas.
It is of course confusing to us that there is no difference between 1 and 60. This happens because the babylonians did not have a concept of 0 – this only came by much later and was quite revolutionary when it did. They did however leave a small extra space after the single nail symbol and the next symbol when the symbol was meant to represent numbers above sixty. This of course did not help in the case where the symbol was exactly 60 as there would be no symbols following it. Because of the lack of a 0 it would have to appear from the context which was meant.
Because of this spacing being relevant however, it makes the babylonian system a positional system where the placement of each symbol is significant to understanding it's meaning.
Roman Numerals
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While Roman numerals are outdated, they are still used today in some specific places: Analogue Clock faces, Chapter numbers and page numbers. They are rarely seen outside of these contexts.
The system is additive, adding up the values of the different symbols. Position is however important, as that tells whether a symbol stands for a number to be added or subtracted. |
Numbers that are meant to be added together are listed with the largest value first, followed by the smaller value(s).
Numbers that are meant to be subtracted are listed with the smallest value first, followed by the larger value(s). Only one lower symbol is allowed in front of a higher symbol. This means that IIV would not be permitted, but would instead have to be written as III.
The system, like most of these early systems, has no symbol for 0, and because of this and the fact that numbers can only be subtracted when they have a larger value to be subtracted from, the system also does not allow for negative numbers.
Furthermore, the system also does not allow any symbol to be repeated more than three times right after each other. In lower numbers this also ensures that the least amount of symbols possible are used.
Numbers that are meant to be subtracted are listed with the smallest value first, followed by the larger value(s). Only one lower symbol is allowed in front of a higher symbol. This means that IIV would not be permitted, but would instead have to be written as III.
The system, like most of these early systems, has no symbol for 0, and because of this and the fact that numbers can only be subtracted when they have a larger value to be subtracted from, the system also does not allow for negative numbers.
Furthermore, the system also does not allow any symbol to be repeated more than three times right after each other. In lower numbers this also ensures that the least amount of symbols possible are used.
The one exception is the number 4 which is sometimes represented by allowing the same symbol to be repeated four times in a row. This is particularly often seen on clocks and watches where the numbers may follow the curve and thereby have the 4 tilted in a way that might be confusing.
Without taking this exception into account however, 1-10 would be written like so: I = 1, II = 2, III = 3, IV = 4, V =5 , VI = 6, VII = 7, VIII = 8, IX = 9, X = 10 |
The Abacus
For learning to use an abacus we'll use a slightly simpler abacus than the one seen above.
An Abacus works by assigning each column of beads a value. For our purpose we will assign the colum furthest to the right the value of ones. The four beads at the bottom then each represent the value 1. The bead at the top represents five. To write six you would then move the top bead down to count 5 (V in roman numerals) and 1 of the four beads up to count 1 (I in roman numerals). The resulting number would be read along the horizontal parting.
The second column from the right counts the tens, so each of the bottom beads represents the number 10 (X in roman numerals) and the top bead represents 50 (L in roman numerals).
It thereby creates an artificial positional system for the roman numerals where each bead corresponds to a roman numeric symbol, which makes math problems far easier to complete.
The second column from the right counts the tens, so each of the bottom beads represents the number 10 (X in roman numerals) and the top bead represents 50 (L in roman numerals).
It thereby creates an artificial positional system for the roman numerals where each bead corresponds to a roman numeric symbol, which makes math problems far easier to complete.
If you wished to add these two together you would start with one of the two numbers. With an abacus of this size you can afford to enter both numbers on other columns that you simply designate as hundreds, tens and ones, so that you can always keep track of them (I've placed these on the left side of the abacus). The one we will be calculating on is the one all the way to the right however. Here, I've chosen DCXXXVII (637) as the starting position (located at the very right of the abacus).
Starting with the hundreds and looking first at the C-beads you will see that in each of the numbers there is a C-bead present, so you push a second C-bead up on the right of your abacus. Then looking at the D-beads you will notice that there are two of those as well, but here we only have one bead available, so we push up one M-bead on the next column and push back the D-bead since it's now gone.
Looking at the next column you will see that each of them has three X-beads, but because that cannot be represented with X beads, we will engage one L-bead and keep one X-bead.
In the final column there are three and two I-beads respectively. That together adds up to one V-bead, so you would pull that down and disengage the I-beads. There of course aren't two V-beads present, therefore we instead engage an X-bead from the next column and disengage the V-bead.
This gives us: MCCLXX (1270)
Looking at the next column you will see that each of them has three X-beads, but because that cannot be represented with X beads, we will engage one L-bead and keep one X-bead.
In the final column there are three and two I-beads respectively. That together adds up to one V-bead, so you would pull that down and disengage the I-beads. There of course aren't two V-beads present, therefore we instead engage an X-bead from the next column and disengage the V-bead.
This gives us: MCCLXX (1270)