48

 

 

48 6x8 2x3x2x2x2

 

The LHS shows 6×8

The RHS shows 2x3X2X2X2  dust ( the 8 is 2x2x2 and the 6 is 2×3 )

As far as the number 48  is concerned the order of rods in the tower is irrelevant, but this needs ‘proving’. Take my word for it at the moment.

48 6x8 2x2x2x2x3

 

So long as the tower is constructed using the rods on the right, the order is irrelevant.

So, as 2x2x2x2x3 is the dust, this means we combine these a pair at a time in any order:

try it yourself..that’s best…but

here’s my mind at work for example:

start with 2, double it double it double it, that’s 16, times 3 is 48 (2 4 8 16 48)

2 threes are six, double it, double it, double it, that’s 48  (6 12 24 48)

2 twos are 4, two fours are 8, three of them is 24, double it, 48

and so on…..

IF YOU HAVE THE TIME AND THE SPACE IN SCHOOL TO DO THIS TILL THE COWS COME HOME AND YOU ARE LITTLE, AND YOU START SLOWLY WITH THE NUMBERS UP TO 10 AT FIRST, STUDYING THE NUMBERS ONE BY ONE FOR A DAY OR TWO EACH FOR EXAMPLE WITHOUT STRESS, YOU WILL ‘GET A FEEL’ FOR THE NUMBER YOU ARE STUDYING WHICH WILL BE VERY POWERFUL IN YOUR FUTURE STUDIES OF THE NUMBER SYSTEM AND OPERATIONS YOU WILL NO DOUBT BE REQUESTED TO CARRY OUT…

(In general, the present school arrangements almost totally inhibit this…)

ps 6×8 is one piece of your ‘tables’, using the dust you see and get the ‘feel’ for 6×8, 8×6, 3×16, 16×3, 2×24, 24×2, 4×12, 12×4, never mind ‘half of 48 is 24’, ‘half of 12 is 6’, ‘half of 48 multiplied by 2 is 48’, ‘a quarter or fourth of 48 is 12’, ‘an eighth of 48 is a half of 12’…and so on till the cows come home…

yap yap yap…

TRY IT

 

 

 

 

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dust lies on top of tables…

SONY DSC

Here’s 8 with its factors: ‘two fours’ and ‘four twos’ which you see to the right.

Remember if you can find rods of the same colour which are the same length as another rod, as in the picture to the left, they are called factors of that number.

At the extreme right is the DUST of 8, ‘two times two times two’, 2x2x2

This is the ATOMIC STRUCTURE OF 8 in terms of multiplication.

Why is it useful and very very good indeed?

Because from the dust, 2x2x2, you can, if you feel like it:

Build ALL combinations of factors of a product

THIS BEATS ‘tables’

DUST EATS ‘tables’

DUST IS ABOVE ‘tables’

DUST BEATS ‘tables’

DUST LIES ON TOP OF ‘tables’ AS WE KNOW ONLY TOO WELL!

ps if you keep saying ‘tables’ it sounds weird too…

dust – your first view of the number 6

Here is  a six rod, with a mat pattern beneath.

SONY DSC

Each line of the pattern can only have rods of the same colour. Only ‘same-coloured’ rods that fit the 6 are allowed this time and for this reason:

These same- coloured rods are FACTORS of 6… (2 and 3 are factors of 6)

6 is called the PRODUCT

The towers to the right say ‘three twos’ and ‘two threes’. They are ‘crossed rods’ and we read them as multiplications. They are equivalent to the line of red rods and the line of green rods to the left.

They are DUST

They are FACTOR TOWERS in their simplest form

They show the ATOMS of 6, if you like…

The INNER STRUCTURE of 6

6 can be re-constructed from this DUST in EVERY (multiplicative) WAY 

The TOWER OF DUST shows the INNER STRUCTURE of the NUMBER 6

(in the multiplicative domain)

going down…millionth, billionth, squillionth

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:

one thousandth 10p-3one millionth 10p-6one billionth 10p-9one trillionth10p-12

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:

1hundredth this is the same as 1over1p2 or it can be 10p-2

They’re different ways of writing down the same thing.

Look here,

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.

going up…million billion godzillion

Here we go…

thousand 10 to the power 3

million10 to the power 6

billion10 to the power 9

trillion10 to the power 12

pattern1?

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

fun with powers of 2

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:

2 to eighth

or:

2 to 8 L 1

or:

2 to the 8th

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

OK

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!

heights of towers and titchy little numbers up and to the right…(logarithms and indices)

OK so this is worth three threes, 3 x 3

3x3

The ‘answer’ or product is 9

Another way of representing the same meaning is:

3x3 log2

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:

3x3x3

3 x 3 x 3, answer is 27

3x3x3 log3

This means the same, 3 x 3 x 3, 27

3x3x3x3

3 x 3 x 3 x 3, answer is 81

3x3x3x3 log 4

This means the same, 3 x 3 x 3 x 3,  81

There is another common way of writing these power towers down:

powers of 3-b

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.

Again:

powers of 3-b

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.