# quite a long one…

‘two thirds of three quarters of four fifths of five halves of two sevenths of seven one-ths of one sixth of six fourths of four thirds of 3 fifths is two fifths’

huatou: learn to read the rods

# two fifths of five sevenths

if the yellow rod is 1, the red is two fifths

if the black rod is 1 the yellow is five sevenths

read straight from the rods: two fifths of five sevenths is two sevenths

the 5 is common to both and cancels out

(the 5 is the ‘denominator’ of the first and ‘numerator’ of the second fractions)

the ‘x’ is an operator, in this case called ‘of’

# the flip law 1

This rule is profound and will change many things, illustrated here with a few rods:

language and rod domain (with a bit of number):

two threes is just as long as three twos, and also they have the same volume

number domain with some signs:

2 x 3 = 3 x 2 = 6

al-jebr domain with signs:

a x b = b x a = c

more and more abstract, more and more general

disambiguation blurb: in the ‘real’ world, two green rods is not the same as three reds. This is why some people object to agreeing that 2 x 3 is the same as 3 x 2. They are correct. However, in the number domain, the answer, which is a pure number, is not affected by the order. The product as a number is invariant to the transformation. If you want another example, it is as invariant as taking a homotopy group functor on the category of topological spaces. You probably don’t need this information. As most people working on calculations are looking for ‘the answer’, one can say that for all intents and purposes, calculationally speaking,

the order of operations in multiplication is irrelevant

further more, if you wish to mention it, and which also makes no difference to the product, the number sentences, transformations or equations, whatever you want to call them, contain the sign ‘x’. This sign is called an operator and it has to be attached to something. It has to be attached either to the first numeral or the second in this case. If it is attached to the first numeral, like this ‘2x’, this ‘whole’ is again called an operator, in this case a ‘doubling’ operator. In language it says ‘two lots’ or ‘two groups’, so this, 2x  3 says

two lots of three or two threes

2x  3

with little children, the easiest and most meaningful form is by using this choice in the attaching of the operator

because:

a) just like in reading, one reads the first numeral first and

b) one doesn’t have to hold the first number in the mind to the same degree as the form below whilst reading the second number. (Saying ‘two threes’ seems somehow less complicated for little learning minds than saying ‘two multiplied by three’)

c) one doesn’t even have to mention the ‘x’ in language, merely recognise it ….two threes

nevertheless the product is still invariant to order

if the operator is attached to the 3, we get 2  x3 which says:

two multiplied by three

2  x3

most teachers call this the ‘correct’ way, but it is just one way

the x3 becomes a ‘trebling’ operator

so, in summary, and for the benefit of little children:

THE ORDER IS IRRELEVANT

and this 2 x 3 with the operator ‘x’ in the middle, at first means ‘two threes’ to little people

later, with much practice, it looks like ‘two threes and three twos’ at the same time

(ps you can introduce all the other ways of saying it whenever you feel it’s appropriate)

JUST UNDERSTAND WHAT’S GOING ON…