0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
T/10 |
E/11 |
NUMBERS. |
Pic by Bing AI (!) My prompt...
(For best results, please view page in landscape).
Quick and easy version...
Note – just before you read this – what's the point?! Well, how would you like to learn a system that makes all basic maths fractions much easier? And at the same time gives you a secret weapon for learning Music theory?! Interested? Read on!
NUMBER BASES.
The thing about numbers
that I would like us to look at first, is “number bases”.
You
already know one, it’s Ten or 10 base -aka; “Decimal”.
This is the most commonly used number base in the world. It is commonly accepted that we humans chose this base for 1 simple reason - we have ten fingers! What if we had 8 or 12 fingers?! Actually – there are number bases other than ten - there are plenty of them – e.g “binary” (2 base) and “hexadecimal (16 base/computer code). Unlike most of us – mathematicians are used to working with alternate number bases -but were only going to look at a few of them...
Later on in the TIME section – we will take a look at 60 base or “Sexagesimal”. You actually already know about this from the clockface! 60 seconds, 60 minutes etc. But for now, I would like to look at something else...You already know about ten base, so there is one particular number base I would like to introduce you to...
Drum roll please ; “Duodecimal” or 12 base! This can also be called “Dozenal” – yes, just as in a dozen eggs! And yes, you are already familiar with this from sets of 5 minutes on the clockface...
START AT 0.
Just before we get started though; counting, doesn't really start from “1”, it starts from “0”!
It’s a small but important point – which becomes very relevant in the music section. Here are 2 examples;
1. You start a running race, after the start of the race, half way round the 1st lap, where are you?
You haven't completed lap 1 yet – you started at 0, not 1 ... 1 is after you have you have completed the 1st lap.
2. You step into a lift at the ground floor. Are you at floor 1? No, you are at floor 0!
And one other point, the 10 digits we use in decimal, start at “0”, and actually finish at “9”! the “10” is where you have started a new cycle...
COMPARISON.
So, I would like to compare the 2 systems side by side...
DUODECIMAL.
What is going on with the X and the E ?! Well, 12 base can only start again at the 12th number –so it needs 2 new numbers.
It would be confusing to call it “Ten” – so they call it “Dec” this ties in with “Decimal”. Also, in old Roman numerals (have you seen them) the symbol for ten was X. As for the E that stands for Eleven and is called “El”.Finally, that “10” at the end, is not TEN! Its called “Do” to avoid confusion!
Quick point – there have been a number of systems recommended for the alternative version of ten and eleven – for example, some people use A and B –
but personally, I think the X and E system does the best job – so I’ll be using them on this site...
What about instead of Zero, we say “Oh”?
Now we have; 0 1 2
3 4 5 6 7 8 9 X E 10
Pronounced; Oh 1 2 3 4 5 6 7 8 9 Dec El
Do... Not too difficult hey?
What is the advantage of 12 base? Well its amazing for Music theory! We’ll look at that on the next page.
Its other advantage is –it makes basic fractions super easy!
FRACTIONS.
Check this out;
Imagine you have 2 pies of the same size. You cut one of them into 10
pieces, and the other into 12 pieces.
You are having a birthday
party, and you don't know how many are coming. You’re also not sure
which pie you will use...
Let’s start with the possible divisions of the 10 pie;
1/2 = 5 pieces each.
1/3
= 3.333 pieces each.
1/4 = 2.5 pieces each.
1/5 = 2 pieces
each.
1/6 = 1.666 pieces each.
Let’s now look at the possible divisions of the 12 pie;
1/2 = 6 pieces each.
1/3
= 4 pieces each.
1/4 = 3 pieces each.
1/5 = 2.4 pieces
each.
1/6 = 2 pieces each.
WHAT’S THE POINT?
Here’s the point, the 5 most basic fractions in order are; 1/2, 1/3, 1/4, 1/5, 1/6,
Out of the 2 pies, Decimal works well with 2 of them... Duodecimal works well with 4 of them!
In other words – if kids learned this in school –it makes basic maths/fractions (drum roll please) MUCH EASIER!!!
Now I confess, it’s too WAY late to change everyone to Duodecimal...However it’s NOT too late to use it as a second - alternative number base.
It’s only similar to learning a second language in school. And the 2nd really interesting thing, is that this knowledge can then be used as a secret weapon/magic key to understanding Music theory! (Explained in detail on the next page).
Earlier, we compared Decimal and Duodecimal – by writing them out in a straight line. What would that look like in a circle? You ready for this?!
CYCLE PIC.
Looks familiar hey?! But with Duodecimal as well as what you're used to on a clock...
To finish this section, here are a few excellent youtube videos about this subject;
From
here, you can;
Go
to the next page.
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back to index page.
Or read on for - More details...
Would you
like to see where I got some of my ideas from?- check out these 2
excellent sites...
Dozenal
society of Great Britain
Dozenal
society of America
A LITTLE HISTORY.
The Romans, counted upwards in blocks of 10. Even though they used a very clunky and basic system – Roman numerals. However, they did realise one thing – that division by 10 was not much good! So, they divided by 12! In fact the origin of the words “inch” and “ounce, comes from a latin word – “Uncia” which means 1/12th! And that is why there are 12 inches in a foot. Also, the original ounce – was divided into 12 not 16.
Roman numerals, were eventually replaced by the Hindu/Arabic numberal system.. They were introduced to Europe by Fibonacci, AKA Leonardo of Pisa - in his 1202 book - Liber abaci. Can you imagine trying to do complex maths using only Roman numerals? Many thanks Leonardo LOL!
Interestingly he is more widely known for the Fibonacci spiral – which I shall look at a little more in the EXTRAS section.
Pic from Wikipedia. Author Romain.
MULTIPLICATION TABLES.
Remember the “multiplication table/square you used to see in school? Well here it is again – but below it, is the same thing in Duodecimal –
hopefully this will help makes things a little clearer;
PRACTICAL EXAMPLES.
Earlier I showed you the very basics of counting in duodecimal.
If you wanted to continue on from here, the next number would be;
11
= “Doz 1” meaning a dozen + 1 = 13 in decimal counting...
Next
up, 12 = Doz 2 = 14 in decimal...
And so on,
Next up, 20 = 2 Doz =2 x 12 = 24 in decimal
etc...
Next up, 21 = 2 doz 1 = (25 in decimal) etc
Finally, in Decimal,12 x 12 = 144 . In Duodecimal, it would be
written like this;10 x 10 =100 – and is pronounced “ 1 Gross”.
There are a few versions of how to continue after this point –but were not interested in that here – because we only need the basics :-)
If you are interested, here is some further reading for you ;
Dozenal society of Great
Britain
Dozenal
society of America
Well, I really hope you found this page interesting?! For now that’s enough info for us to be able to move on to the next step; the MUSIC page!
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to the next page.
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back to index page.