On Fractions

I think these are great: more adaptable and more accurate than decimals.

But it's the division thing which always bugs me. You know: "turn the divisor upside down and multiply" - what the hell? Well, ok, I'll do it....

Well, doodling during a lesson the other day (yes I know I am supposed to be the teacher), I worked out a kind of rationalisation.

Take 3/6 divided by 1/2.

Flip over the 1/2 to make 2.

3/6 x 2 = 6/6 = 1.

So a half goes into 3/6 once. Makes sense, as 3/6 is a half.


But think of it without fractions.

6 / 3 = 2, right?

but 6 x 1/3 also = 2.

Because division is the inverse of multiplication, you are basically just doing the inverse. Also because multiplication is a scaling process, when you times by a number less than 1, the answer will be less than the number you were trying to scale. It scales downwards. 4x2 = 8 but 4x1/2 = 2.


And reciprocals are essential here. A reciprocal is any number you multiply any other number by to make 1. In practice this means that the reciprocal of a whole number is 1/that number.

1/6 x 6 = 1

But the reciprocal of any fraction is that fraction turned upside down.


3/4 - 4/3.

This is because you can imagine any whole number as n/1.

I guess you could say the reciprocal of a number is the inverse of that number...

So, 10/2 = 5. Ie 2 goes into 10 five times.

And 10 x 1/2 = 5. ie 10 lots of 1/2 make 5.

And 10 / 1/2 = 10 x2 = 20. ie a half goes into 10 20 times, which is the same as multiplying ten by two.

The fact that this holds makes it a lot easier to divide fractions, so it is a trick, but a useful one. There are proper proofs of it, but this isn't supposed to be a proof, only a sort of meditation on the subject.

On The Importance Of Factorising

Without it, even linear equations become extremely difficult to re-arrange.

Take 6xy + 3x - 2y = 7

Re-arrange this for x.

MOVE the non x bit.

6xy + 3x = 7- 2y.

SPOT the common factor

x(6y + 3) = 7 - 2y.

DIVIDE by (6y+3)

x = 7-2y
       ----
       6y+3

PIECE of the proverbial.

Indices And Differentiation

Yesterday, courtesy of my beautiful and awesomely intelligent maths tutor, I suddenly worked out a way of making differentiation with negative indices slightly easier.

When inputting the x value to find the gradient at that point, to make the final stage of adding it all up much easier, turn your x^-2 into 1/x². It makes calculations a lot, lot easier.

Laws of Indices

Are pretty simple.

a^m x aⁿ = a^m+n

a^m / aⁿ = a ^m-n

(a^m)ⁿ = a ^mxn


But a^m + aⁿ DOES NOT equal anything other than what it says.
There isn't much you can do to collect the terms if the indices are different, or if the constant is different (ie x^m + y^m)

a^m + a^m = 2(a^m), of course.

Fractional Indices (2)

Well yes it gets a bit harder.

You see...a ^n/m


means ^m√aⁿ


...but I will try to explain this later.

Fractional Indices (1)

Think about it.


Think about it.


x is not just x. x is x¹. x is x raised to the power of one: x multiplied by itself no times at all; x just being x. So then x ¹ is x on its own. x is, therefore, x.

So then, x^1/2 is going to be a number that, multiplied by itself, makes x.

In other words, the square root of x.


√x = x ^1/2


It does get a bit more complicated than this....

Negative Indices

Negative Indices

Now these seem simple enough: you raise a number to a power and it usually means multiplying a number by itself n times.

a³ = a x a x a

All numbers to the power of 1 are themselves, and all numbers to the power 0 are one.

a⁰ = 1



Easy enough.


But then you can also have negative powers.

a^-3

What? You can’t multiply a number by itself a negative number of times!

Well, no, clearly. But you can see how negative powers come about.


It’s to do with the powers rule.

aⁿ x a^m = a^n+m

You can see this if you write out aⁿ and a^m in full.

aⁿ x a^-n = a^{n+ -n}

= a^n-n
=a¹
=a

Therefore, the negative powers are used to denote reciprocals (the number you multiply n by to get 1 – so 1/6 is the reciprocal of 6 – and it is always 1/n.

So.


3³ =27

3^-3 = 1/27 (or 1/33 – the reciprocal).


Negative powers are easy to manipulate, they just seem a bit weird until you think that positive powers are going up by multiplying the number by itself:

2⁴ is 2 x 2 x 2 x 2

But if you go down, towards 0, the same process is division by two.

64...32...16..8...4..2..

This continues as you go down below zero:

......2, 1, ½, ¼, 1/8, 1/16......

(2¹, 2⁰, 2^-1, 2^-2, 2^-3, 2^-4)

And you can see that these are reciprocals of the powers of 2.

The same applies for the powers of each number.

C1 and S1 taken!

Apologies for the lack of posting. I sort of became involved in my actual revision!

But I took C1 and S1 on Friday. C1 was ok but S1 was really tricky. Not sure how I did there.

Anyhow. I'm doing C2 now but I will try to post some more pure-related posts. I'm not sure I want to think about stats again!