keep making the square bigger and look for patterns we’ve started above….
I can’t do any more of this ‘tables’ stuff, but I am asked about the issue so much…
Look, here’s a ‘tables’ square, slightly adjusted:
I’ve removed the 1x and 10x sectors. 1x is trivial, 10x needs special treatment.
If you fully appreciate the flip rule you can forget the grey airbrushed section.
The square numbers in the blue-green squares are a special beautiful group, well worth studying. Many patterns and much al-jebr live here…
The rest are in 8 columns, from 1 to 8. Add up the numbers from 1 to 8 in your head and it’s 36. [I did 8×9 and halved it).
The orange numbers can be found by doubling from single digit numbers. (Conversely by halving from the products).
The blue by treblings from single digit numbers.
There are 4 spaces left, 5×6, 5×7, 6×7, 6×9.
double 15, the two primes, 5 and 7 make 35, double 21 and double 27.
Study George Cuisenaire’s original product wall chart:
You can see all the doublings. Look carefully.
These products, the numbers in black, form
MILESTONES in the unknown territory to 100
By studying them as numbers, as we have previously discussed, their inner structures will become apparent, thus lessening the awful stress on memory so prevalent in today’s schools.
In addition by briefly studying the Primes to 100 it will become apparent to the little yellow belts that many numbers are rich in factors and many are not. They will generate a ‘feel’ for the rich numbers, learn their inner structures by familiarity and learn, as an aside, the so called ‘tables’ relevant to that number.
This however takes
A CHANGE OF PROGRAMMING
even in the minds of teachers, never mind the administrators
THIS IS PROBLEMATIC…
because even you and I dear reader
IMAGINE OURSELVES TO BE FREE
huatou: ‘Am I free?’
Here we go….however just before we begin the descent, I would like to claim that as far as I know, the following kind of representation has not been done before. I stress ‘as far as I know’, so apologies to anyone who has. I may be up for a Nobel Prize…whoa!
Instead of the rod power towers going up, they could of course go down. They’d be subterranean establishments like secret government underground levels where they interrogate aliens and hide their nuclear mission control nodes.
Towers that ‘go down’ are a bit hard to do with real rods unless you had good glue and worked underneath the table, however they can be represented if we use the standard L form for towers layed down flat on a surface then go the other way like this:
10 to the minus 12th, that’s how you say it.
The minus sign is a bit upsetting, but just think of it as making the power towers go down. The other way to think about this is to understand that its a short hand way of writing down the fractions, one thousandth, one millionth etc. i.e. it is one whole one reduced in size by a thousandth or a millionth. Another way is to understand that its one whole divided by a hundred, a thousand or a million. Look, here’s one hundredth:
They’re different ways of writing down the same thing.
for millionth you say micro, for example you could say a micrometre – an anthrax spore is about 6 micrometres across.
for billionth you say nano, for example a nanometre, a billionth of a metre – the diameter of a ribosome is about 20 nanometres.
for trillionth you say pico, you could say 6 picometres, 6 trillionths of a metre – an average atom is about 100 picometres in diameter.
Then you get, femto, atto, zepto and yocto.
One yoctometre is 10 raised to the power -24
A squillionth of an inch is half a grain of spacetime.
Here we go…
A 3 rod naturally, as THE COMMON DIFFERENCE in this particular staircase is 3
Just for fun here’s the names of some of the other big guys:
quadrillion 10 to the 15th
quintillion 10 to the 18th
sextillion 10 to the 21st power
septillion 10 the 24th
octillion, nonillion, decillion, undecillion, dudecillion, tredecillion, quatuordecillion, quindeciliion, sexdecillion, septendecillion, octodecillion, novemdecillion, vigintilliion…and so on for ever…
Of Latin origin except for:
godzillion 10 to the power of godzilla
Computers work with base 2 because switches can be either switched on or switched off just like your light switches. ON or OFF, that’s 2, so its base two. (We have 10 things on the ends of our two hands so we count in base 10, its as easy as that. If we had 8 fingers like the Martians we’d count in base 8 obviously).
Here is 2 raised to the power 8:
If you work out 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 you get 256
THIS IS CALLED ONE BYTE, there’s 8 BITS in a byte. That’s the INDEX, 8 above.
Here’s a typical BYTE: 1 0 1 0 1 1 0 1
Its a number in base 2, where you can only have the numbers 0 (which represents ‘OFF’) and 1 (which represents ‘ON’).
The different columns are similar (in fact isomorphic is the word, I love that word) to our normal (base 10) number system, but every column is worth twice as much as the one before, unlike base 10 where every column is worth 10 times more than the one before.
I just wanted to tell you about multiples of bytes, so here goes:
1 byte is 2 to the power 8
1 kilobyte is 2 to the power 10
1 megabyte is 2 to the power 20
1 gigabyte is 2 to the power 30
1 terabyte is 2 to the power 40
1 petabyte is 2 to the power 50
1 exabyte is 2 to the power 60
1 zettabyte is 2 to the power 70
1 yottabyte is 2 to the power 80
1 yottabyte is enough memory to store 250 trillion DVD’s
and remember, that’s a pile of 80 crossed red rods if you call the red rod 2. I bet you can’t make one of them… However, is there another way using rods…? I’m not telling…
One of the most powerful computers at the moment is TITAN, built by Cray at Oak Ridge, and it can only store 17.59 terrabytes of information. How pathetic is that? (Got to admit, it’s pretty fast though….)
However the US government has a computer called DAWN which can simulate the brain of a rat, that’s coming on…. In another 20 years or so, there will be computers that can simulate the brain of a human….that’s SCARY and we’d better WATCH OUT!
OK so this is worth three threes, 3 x 3
The ‘answer’ or product is 9
Another way of representing the same meaning is:
It means 3 raised to the power 2
The height of the tower is 2.
Next its 3 x 3 x 3 followed by 3 x 3 x 3 x 3:
3 x 3 x 3, answer is 27
This means the same, 3 x 3 x 3, 27
3 x 3 x 3 x 3, answer is 81
This means the same, 3 x 3 x 3 x 3, 81
There is another common way of writing these power towers down:
The little number is THE HEIGHT OF THE TOWER, called the INDEX
Just as the names of the numbers after ten have unusual names like ‘eleven’ and twelve’, the first two of these have other common names. Numbers raised to the power 2 can be called ‘squared’, i.e. 3 squared, and numbers raised to the power 3 can be called ‘cubed’. After that its all ‘regular’, they are called something ‘raised to the power something’, such as 3 raised to the power 4 in the last example.
( You can easily make a square using three threes on the flat. Multiply the length of 2 sides together and you get 9, the same as 3 x 3 . You can make a cube of three rods, then if you multiply the 3 sides together you get 3 x 3 x 3 which is 27.
By the way, the square root of 9 is 3, and the cube root of 27 is 3…its all playing in the same ball park).
10 raised to the power 100 is called googol.
The number of atoms in the universe is about 10 raised to the power 79, so googol is far greater than the number of atoms in the whole universe.
So the INDICES are the little numbers. Now wait for it,
THEY ARE ALSO THE LOGARITHMS OF THE ANSWER, provided you are talking about the same colour rods, in this case 3 rods or light green:
2 is the LOGARITHM of 9 in BASE 3
3 is the LOGARITHM of 27 in BASE 3
4 is the LOGARITHM of 81 in BASE 3
The main use for these logarithms in the olden days of torture in school was that:
INSTEAD OF MULTIPLYING TWO NUMBERS TOGETHER YOU CAN ADD THEIR LOGARITHMS.
The same is true of indices, and these days this is still useful with very large or small numbers, as in maths, physics and engineering etc.
I’m getting a bit fed up with this now, so just let’s say to finish and for a quick example, 3 squared multiplied by 3 raised to the power 200 is 3 raised to the power 202. As to the answer, I’m too tired to figure it out. Goodbye.
Logarithms sound bad I know, but they’re not.
Say, ‘ten cross ten’. The answer is 100 is it not? It is 2 layers high:
This means the same thing, but we use less rods when the towers get higher. We can turn this one on its side if we like:
This means 10×10, 10 squared, 10 raised to the power 2, answer is 100
THE LOGARITHM OF 100 IS 2
…if you are using orange rods and you call an orange rod 10
THE LOGARITHM IS THE HEIGHT OF THE TOWER, THAT’S ALL…………….
The only thing to stress is that the tower must be made of the same colour of rods.
This colour is called the BASE