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# Basic Maths

Basic Maths

• ### Is it worth contacting HMRC to warn of VAT refund?

Simple question, I think.

Suppose clients fee was £650 20 years ago, and practice inflation is 4%.

My naive assumption was this was 650*0.04^20 in excel.  But its seems not, as I get 7.146825580544E-26 - oops!

Sure I can do it the long way over 20 rows and 2 columns and arrive at £1,369, but there must be a more elegant way.

### Replies

By mrme89
20th Feb 2013 10:44

=650*(1+4%)^20

=650*(1+4%)^20

Kind Regards

John

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20th Feb 2013 10:45

=650*1.04^19

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20th Feb 2013 11:00

^ This

19james88 wrote:

=650*1.04^19

^ This

Though the other post would work too.

Your formula was reducing rather than increasing!

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20th Feb 2013 23:22

Polynomial expansion

And as we all know, for small x, where x >> x^2,

(1 + x) ^ n ~= 1 + nx = 1 + (20 * 0.04) = 1.8 !   Edit:Ooops sorry about appalling coffee-break arithmetic !

Hence approximately the value is 1.8 * 650 = 1170.. obviously x is not small enough ... :-(

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20th Feb 2013 14:56

Err NO

(20*0.04) =0.8

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20th Feb 2013 15:04

Bonkers maths

This is my favourite piece of bonkers maths

a = b

so          a2 = ab

so          a2 – b2 = ab – b2

so          aa - bb = ab - bb

so         (a +b)(a – b) = b(a – b)

so          a + b = b

so         a + a = a

so         2a = a

so          2 = 1

There is of course a flaw in this. But can anyone spot it?

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20th Feb 2013 15:12

That one has whiskers on it

The move from line 4 to line 5 of course

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20th Feb 2013 16:05

Hmm

Chris Smail wrote:

The move from line 4 to line 5 of course

That's where I looked hard too, but surely dividing both sides by (a-b) would give the odd result on line 5?

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20th Feb 2013 16:01

It may have whiskers

but I don't think I'd seen it before. Good one!

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20th Feb 2013 16:03

Another useful approximation is

70/interest rate pa gives the number of years needed to double your money. Obviously this can be extended to non-financial items as well.

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20th Feb 2013 16:11

Is it...

... because you can't alter both sides while brackets are involved?  So you can't do what I thought you could and divide both sides by (a - b)?

Curious now...

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20th Feb 2013 16:15

It's because

(a-b)=0 and so you are dividing by zero

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20th Feb 2013 16:32

But

paulwakefield1 wrote:

(a-b)=0 and so you are dividing by zero

But the same zero term, so surely that can't be the only reason?  If you did a mathamatical equation that had a term that happened to be zero, that wouldn't stop you from rearranging it would it?

Meh, I think this has gone beyond my comprehension level, I won't embaress myself by asking any more questions :)

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20th Feb 2013 16:38

@CC

You're looking at line 5. Line 1 is a = b

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22nd Feb 2013 10:39

And I thought you lot were numerate

Remember a = b so a-b = 0

hence (a +b)(a – b) = b(a – b) works because anything divided by 0 is zero

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20th Feb 2013 17:36

Oh well...

... that's not then where the problem lies.

The problem is in the preceding line.

You can't get from a2 - b2 = ab - b2 to aa - bb = ab -bb, unless a = b = 2. Which clearly isn't the case, given the conclusion.

Anything divided by 0 is infinity.

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By Triggle
20th Feb 2013 17:50

Anything divided by zero is an impossible result - not zero EDIT: nor infinity

Try in on a calculator and you get the E and the keyboard stops working

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20th Feb 2013 17:50

1) To find a woman you need time and money,

Therefore:

Woman = Time x Money

2) Time is money,

So:

Time = Money

3) Therefore:

Woman = Money x Money

Or:

Woman = (Money)²

4) Money is the root of all problems,

So:

Money = √(Problems)

5) Therefore:

Woman = (√(Problems))²

Or simply:

Woman = Problems

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20th Feb 2013 17:53

@Triggle

If you divide something by zero without your precious calculator, the result is infinitely large (to state it precisely), represented by the mathematical symbol ∞.

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20th Feb 2013 17:55

@George

I think the formula is meant to read "a squared" rather than an odd way of writing 2 x a in which case the step works.

As said, it's the dividing by zero that blows it  .

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By Triggle
20th Feb 2013 17:58

No the result is in fact any number under thge Sun. It does not compute mathematically.

1800 / 0 = 0

Because 0 x 1800 = 0

24.28/0 = 0

Because 24.28 x 0 = 0

Try it with all the numbers you can think of (including imaginary number and zero itself). You'll be there till infinity though.

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20th Feb 2013 18:12

You can't prove...

... what the result of the division is by then multiplying by 0, and I thought you said that something divided by zero wasn't zero and now you're saying it is.

Any number can be divided by zero an infintely large number of times. It isn't 0, it isn't 1, it isn't 2, it isn't 3... it isn't 24.28... it isn't 1800, it isn't 1801...

Thanks Paul. I'm now as confused as Constantly, because, in my mind as well, despite that the two terms should still cancel.

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20th Feb 2013 18:16

They do

From the 2nd "So" to the 5th "So" are just different ways of writing 0 = 0 and the steps are all perfectly valid. But when you divide through by (a-b), you are dividing by nought and it all falls apart.

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By Triggle
20th Feb 2013 18:21

Any number divided by zero is any number and all numbers including 0 simultaneously.

There are infinite answers to the same calculation which is not the same as saying the result is infinity. The resut can infinitely small, infinitely large or anywhere in between.

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By BKD
20th Feb 2013 20:06

x/0 has no value whatsoever, because there is NO number (at least not in the real world, where most of us live) which when multiplied by 0 will give you x.

The expression is therefore undefined and has no meaning. Trying to prove that it has a value - be that a single value or an infinite number of values - is pointless. (Though I'm guessing that there is some theoretical mathematician out there that is trying to prove me wrong.)

If you do want to prove it to yourselves, try plotting a graph of y=1/x. You'll find that the closer and closer x gets to 0, the closer y gets to infinity. But, and this is the important bit, the curve never touches the y-axis, which it would have to do in order to give a value to 1/0. So x/0 can never, ever, have any  value. QED

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20th Feb 2013 20:38

Surely

Because the line never touches the axis then x/0 = infinity.

a separate note if you were to rotate this line around the x axis then the cone shape you make would have an infinite volume but a finite surface area. love maths

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By BKD
20th Feb 2013 21:02

What is infinity?

If that were so, it is still the case that infinity has no value, therefore x/0 has no value. Theoretically, infinity is a point somewhere in the far, far distance on (at the end of) the y-axis. But because the 1/x curve never touches the y-axis. x/0 can never be infinity either.

And for the same reason the cone has an infinite surface area as well as infinite volume - the base of the cone gets wider and wider as x approaches 0 but never stops (it would stop only when it touched the y-axis at that theoretical point called infinity. Which would paradoxically result in a finite volume and finite surface area.)

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20th Feb 2013 22:20

Just realised an error...

... in my formulae.

At step 4, it should be money is the root of all evil.

x/0 = ∞ by the way, where ∞ isn't infinity, but an infinitely large number! Ask a mathematician!

The only way that x/0 couldn't be a number BKD is if x or 0 weren't numbers. :)

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By BKD
20th Feb 2013 22:46

Let's assume that ...

...  ∞ does in fact represent a number, albeit infinitely large. So, tell me, what is the result of ∞ x 0? There should be no need to ask a mathematician - it doesn't matter how big ∞ is, the answer has to be 0. That being the case, x/0 cannot be ∞.

A mathematician will tell you that x/0 is undefined - it has no value.

I spent several years at university doing proof theory. Since I've managed to prove that x/0 cannot be ∞, the onus is now on you to prove that it is.

And for the pedants amongst us, "The love of money is the root of all evil."

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20th Feb 2013 23:25

Divide by zero

If you divide by zero all bets are off.

And results incorrect.

eg.

2 x 0 = 3 x 0

divide by zero on both sides ...

2  =  3

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21st Feb 2013 00:08

BKD

Saying you have proved your point then quoting your GCSE in maths or whatever does not mean your point is proven. Not to mention the fact that you missed my very interesting point earlier.

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By BKD
21st Feb 2013 08:12

Proof

Pottedbeef wrote:

Saying you have proved your point then quoting your GCSE in maths or whatever does not mean your point is proven. Not to mention the fact that you missed my very interesting point earlier.

You've misunderstood the meaning of 'proof' in proof theory. I have demonstrated using logical argument that x/0 cannot equal infinity (or, to be precise, an infinitely large number). All I've done is invite others to counter that with similar logical argument. Simply saying that x and 0 are both numbers doesn't count. (But if one persists witht that argument, there is a logical counter to it.)

As for your 'interesting' point, which one? Or did you miss my response at 21:02 yesterday?

And as for Wikipedia, that doesn't surprise me. My argument is the most basic and well-known proof of why x/0 does not compute.

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21st Feb 2013 00:15

Also

your last point looks scarily like the wikipedia entry for x/0

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21st Feb 2013 09:09

!

Also, when I searched for 'divide by zero' I came across:

:)

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By neileg
21st Feb 2013 09:33

Linear analysis

If you really want to understand the maths around infinity and dividing by zero, get yourselves a good book on linear analysis. You'll find that lots of apparently sound functions have a discontinuity where a graph would cross one of the axes. Mathematicians can spend their entire career studying these discontinuities.

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By kdauda
21st Feb 2013 09:47

I'm with BKD

Also did maths at university. I'm with BKD :)

In the real world (that most of us live in!) something divided by zero is undefined - it has no meaning. You could call it infinity or whatever else you want, but that something would not be a number like all the other 'real' numbers so you can't do the usual arithmetic (add, multiply, divide, subtract) with it. When we try to do arithmetic with it, we end up with those odd results.

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By BKD
21st Feb 2013 10:17

Moving on ...

...

If  2/9 is 0.22222222....., 7/9 is 0.77777777......., etc why is 9/9 not 0.99999999........?

(It's a rhetorical question - I know the answer)

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21st Feb 2013 10:32

:)

BKD wrote:

...

If  2/9 is 0.22222222....., 7/9 is 0.77777777......., etc why is 9/9 not 0.99999999........?

(It's a rhetorical question - I know the answer)

I like that one, mainly because my little brain can grasp the concept :)

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21st Feb 2013 10:19

1/0 = Infinity

I am with George here assuming that you are not dividing 0/0

1/0 = Infinity.

Here is why:

1 / 0.5 =    1

1/ 0.3333 = 3

1/0.25 = 4

Therefore if you continue dividing by smaller and smaller fractions then the answer gets bigger and bigger. Ultimately therefore it follows that dividing 1 by zero would yield an infinite number.

Therefore 1/0 = Infinity

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21st Feb 2013 10:33

No, no, no BKD (and others)

What you've proven is that you have a closed mind and aren't willing to accept anything outside your own understanding.

You, Triggle and Kdauda are confusing "undefined" with "I don't know the answer to that one, my calculator won't do it, and I don't do abstract concepts".

You are correct that ∞ x 0 = 0, but that's just one of the possible values, as I'll show you.

The problem, as has been identified, is that if you introduce a division by zero into an equality, the equality breaks. We don't want an equality to break; equality's very important, particularly in the workplace.

What we need is an expression to replace the division by zero, so that the equality doesn't break.

Since you people that "did maths at university" will, I am sure, be comfortable with the fact that:

1 ÷ 0.01 = 100, and

1 ÷ 0.001 = 1,000,

and so on, such that the more you tend towards zero the larger the result gets.

It follows that when you reach zero (which is what you do when you get to 1 ÷ 0, which is what we're talking about), the number is incomprehensibly large; it's larger than any number we know; there can't be any larger number.

Let's call this incomrehensibly large number (of which there can't be any larger number) ∞ for arguments sake. It's the inverse of 0.

Now because there cant be any larger number than ∞, it follows that:

2 x ∞ = ∞, and

3 x ∞ = ∞, and

4 x ∞ = ∞,

and so on.

Already ∞ breaks the conventional rules of multiplication. So why should those rules not also break when you multiply ∞ by zero.

We already know that ∞ can be expressed in many forms using numbers that we do know:

0/0, 1/0, 2/0, 3/0, and so on until we reach ∞/0.

When we multiply by zero, the denominator in all the above fractions, we're left with the numerators, 0, 1, 2, 3, and so on until we get to ∞.

So we must conclude, must we not, that 0 ≤ (0 x ∞) ≤ ∞.

So when mikewhit says:

2 x 0 = 3 x 0 and the says "now divide by zero", we can all shout "you can't divide by zero, but you could multiply by the inverse of zero!".

So then, instead of, incorrectly getting to 2 = 3 as in Andrew's bonkers equality,

we move to 2 x 0 x ∞ = 3 x 0 x ∞,

which can be simplfied to ∞ = ∞.

And there we go, now our equality works!

Phew! ∞ saved the day!

Quod Erat Demonstrandum.

Live long and prosper!

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21st Feb 2013 10:39

George Attazder wrote:

2 x ∞ = ∞, and

3 x ∞ = ∞, and

4 x ∞ = ∞,

I remember my physics teacher astounding the class by telling us there are many infinities, as there are an infinite number of even numbers, and even numbers represent half of all numbers, so there is a number twice of infinity (which is also conveniently called infinity).  That about sums up what I learned in physics.  Oh, and motorbikes don't generate enough grip to stay on the road (he proved this on the board knowing full well the method of transport I was using to go home), and yet they somehow do!  I've since been told countless times never to try and prove a motorcycle can lean at high speeds and maintain grip, nor that a plane can actually stay in the air.

Btw, does anyone remember when this topic was about correct a simple mistake in an Excel forumula?

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By markfd
23rd Feb 2013 09:18

Actual maths...

George Attazder wrote:

Since you people that "did maths at university" will, I am sure, be comfortable with the fact that:

1 ÷ 0.01 = 100, and

1 ÷ 0.001 = 1,000,

and so on, such that the more you tend towards zero the larger the result gets.

It follows that when you reach zero (which is what you do when you get to 1 ÷ 0, which is what we're talking about), the number is incomprehensibly large; it's larger than any number we know; there can't be any larger number.

Unfortunately the problem is that you could argue similarly with -0.01, -0.001, and so on, whereby the numbers get smaller and smaller on their way to - infinity.

Hence there isn't a uniquely defined value for 1/0.  The value can't depend on which direction you approach zero from.

There are number systems where this works eg in spherical geometry 0 = 1/infinity, to visualise this you can think of the North pole as infinity and the South pole as zero.

The other problem is that your statement "there can't be any larger number" is actually wrong.  There are several sizes of infinity (and indeed several ways of comparing them).  I'm sure Wikepedia explains this if you look up power sets, aleph 0, and the continuum hypothesis.  That's why one can prove there are more 'real' numbers than 'rationals' using an argument known as Cantor's diagonal slash.

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By BKD
21st Feb 2013 10:27

Or ...

1 = 1

1 x 1 = -1 x -1

√(1 x 1) = √(-1 x -1)

√1 x √1 = √-1 x √-1

1 = -1

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21st Feb 2013 10:31

No BKD

The last line should be:

1 = i²

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By BKD
21st Feb 2013 10:57

Exactly, George

George Attazder wrote:

The last line should be:

1 = i²

The fallacy works only because one has to use imaginary numbers. Actually, your suggestion could easily be inserted as a second-last line.

My point being that one can prove anything one wants by introducing non-real factors into the argument.

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By kdauda
21st Feb 2013 10:41

'Real' world

@ George

Sort of agree with you but I did say in the 'real' world. I know I didn't define what I meant by the 'real' world (haven't got time), but one condition for the real world would certainly be that you have to be able to do conventional multiplication. But what you were doing is not in that 'real' world that I was referring to. Actually you can do arithmetic with different levels of infinities... again, not in the 'real' world.

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By Triggle
21st Feb 2013 10:48

The square root of -1?

Now there's a concept.

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By BKD
21st Feb 2013 13:48

No, no ,no, no George

What we need is an expression to replace the division by zero, so that the equality doesn't break.

That, I think you will find, is called moving the goalposts. As one gets closer and closer to zero, the result does become infinitely greater and greater. But we're not talking about the result when the denominator is as close to zero as we can get. We're talking about the result when it is zero. 0 may be a number, but it is a number with special properties, used to denote null, nil, nothing. 0 is not the inverse of ∞, because 0 is finite. The inverse of ∞ is 1/∞. An infinitesimally small number - close to zero, but not zero.

As Neil says, you can get as close to the y-axis as you want - but as soon as you touch it, the whole thing falls apart.

∞ x 0 can only ever have one result, not because of the properties of  ∞, but because of the properties of 0. ∞ doesn't have any special numerical property - it is just a number tht happens to be so large that no-one can write it down.

So why should those rules not break when you multiply ∞ by zero

Because they don't. Although I accept that your 'proof' is elegant, it is based on fallacies. For instance, If we assume that ∞ is the largest number there is, there is no such thing as 2 x ∞, 3 x ∞ etc. But ∞ x 0 does have a value, a single value.

Also - When we multiply by zero, the denominator in all the above fractions, we're left with the numerators, 0, 1, 2, 3, and so on until we get to ∞.

Er... no we're not. Multiply anything by zero and we're left with zero. You're argument works only if zero is treated the same as any other number (including ). But zero is not the same as any other number.

So we each have our 'proof'. I'll defer to the real mathematicians, though. I do not know a single one, including those that I studied with, that when asked the question "what does x/0 equal?" will respond with anything other than "it doesn't equal anything".

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21st Feb 2013 11:03

It's not my fault...

... if you're averse to abstract concepts! :)

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By mwngiol
21st Feb 2013 11:10

Gosh

So this is basic maths is it? Blimey!

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