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We've found a real utterly nutty PURE MATHEMATICIAN! Regular visitors to the site will know this character... but we had no idea she really exists AND she has even found a useful role to play in the community by being a maths teacher! Big respect and thanks to the wonderful Pam Garnett of Wolfreton School for letting us use her photo!

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Welcome! This is where our Pure Mathematicians experiment on little bits of maths and try to answer some big questions. 6lt6i

If there's a question that YOU would like us to investigate, then please us through the mailroom. If we think it's interesting* enough and we can come up with an answer then you might see it here... eventually! 26t6n

* HINT: Don't ask us if 1 is a prime number. Been there, done that, got nowhere, gave up, came home and put the kettle on.
* Also please DON'T ask us to do long sums that you could do yourself on a calculator!

GUARANTEE: You may rest assured that no numbers or sums are harmed during the course of our investigations. 6d2x14

Here are the answers to some of the questions that we've already been asked. 201v2e

WHAT IS THE AREA OF AN ANULUS (or RING)? asked (and answered) by Alex Jeffreys 6f3d2k

First of all we better see what an anulus is. It's the posh name for the red bit of this picture - so it's like a disc with a hole in it, or a washer or a ring or a squashed doughnut. 6j2q5m

Obviously you can work out the area by subtracting the area of the little circle from the area of the big circle, and to work out these areas you need to know the radii of both circles.

HOWEVER... it's possible to get the area of the red bit with just one measurement! We thought you might like to think about how to do this before you look at... the area of an anulus answer!

WHAT IS THE SQUARE ROOT OF i? asked by Phil M. of Manchester 545q11

In Numbers - the Key to the Universe we meet the very spooky imaginary number "i", which is the square root of -1. Therefore you can write: i2 = -1 5s2kr

Understanding all about i is tough enough, but just to be awkward Phil asked us, what is the square root of i? 2bs3m

	It took a while but we think we've got it: 1z6l12
	And what's more... we can prove it! If you square it, you get i. Look: 5l6u2v

What is the cube root of i? 2v3s3h

We had the square root of i on our site for ages, but LOTS of people demanded to know what is the cube root of i. Luckily that's not too ugly as you'll see here. 6l3sh

There are actually two more cube roots of i, and one of them is just -i!
If you work out (-i)³ you get (-i)×(-i)×(-i) = -(i×i×i) = -(-1×i) = i.
Thanks to GOSHKO for telling us this!

Then it got silly. People then demanded to know what is the FOURTH root of i. Here it is and it serves you right...

And yes, we were even asked for the fifth root of i. There are a few answers to this (including 0.951056516 + 0.309016994i) but because i⁵ = i there is one very simple answer: i.

While we're dealing with "i" here are some more questions we were asked: 19qs

What is the square root of -i ? asked by Kitty Soft 2v2l1t

You just multiply the answer for the square root of i (see above) by i. You get SQRT(-i) = (i-1)/SQRT2 s534r

If i2=-1 then what does -i2=? asked by Alex Jeffreys 58281y

You've got to be careful here. Usually -i2 would mean (-i)2 and we think that ALSO makes -1. 4dk66

However if you're not squaring the "-" then you're working out -(i)² and that makes +1.

What does ii=? asked by loads of people g5t4q

This is very strange. The answer is a real number: 0.207879576... 4s3e4s

What makes it even stranger is that there are a lot more answers. This all comes from Euler's Formula, so guess what the next question is? 4j6a47

WHAT IS EULER'S FORMULA? e3g1i

Here it is: eiθ = cosθ + i sinθ 4a283o

To understand this you need to know about e and angles measured in radians, which means that an angle of π = 180º .
The special case where θ = π gives us e^iπ = cosπ + i sinπ but cosπ = -1 and sinπ = 0 so we end up with the utterly lovely equation ...

Euler's Identity: e^iπ = -1

WHAT IS THE AREA OF AN ELLIPSE? first asked by Jake Chan 1z546j

In The Area of an Ellipse. 66g5l

WHAT IS AN ARCTAN? asked by Jordan Watts 5o43u

In "Do You Feel Lucky" it says that you can calculate PI using the fomrula 7111

PI = 4[ARCTAN(1/2)+ARCTAN(1/3)]. Easy! (You have to work this out in radians by the way, not degrees.)

ARCTAN is the inverse of TAN, which comes up in trigonometry and is a way of turning fractions into angles. If you put TAN 45^o into a calculator you should get the answer 1. Or, you can put this the other way round, if you put the inverse of TAN&nbsp1 (or SHIFT TAN&nbsp1) into a calculator you get 45^o. You can also say the ARCTAN of 1 is 45^o.

You can also have ARCSIN and ARCCOS. E.g. SIN 30^o=0.5 so ARCSIN 0.5=30^o

HOW CAN I FIND THE CENTRE OF A CIRCLE? asked by Mello 563x44

There are several ways, but our favourite is the USING A BOOK method! 5b6p18

HOW DO YOU WORK OUT SQUARE ROOTS ONLY USING A PENCIL AND PAPER? asked by Alistair Lynn 4b3p4r

We've had several answers to this, and now you'll find them on a separate page: 4x1v25

How to work out Square Roots without a calculator

WHAT IS THE VOLUME OF A CONE (or a pyramid)? asked by Aryan S. and others 13e6c

The volume of a cone, or any other pyramid shaped object is: 1/3 x base area x perpendicular height These two doodles show the volumes of a cone and a rectangular based pyramid. 2mc3o

Notice that the point of the pyramid (called "the apex") doesn't have to be directly above the middle of the base. It can be leaning over a bit if you want, just so long as you measure the perpendicular height correctly.

If the green sides all measure "1" then the blue sides measure sqroot2 and the red sides measure sqroot3.

HOW CAN I WORK OUT THE nth TERM OF THE FIBONACCI SERIES? asked by Seung Yun Jee 534d2r

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To get each number in the Fibonacci series you have to add the last two numbers together - but suppose you want to know the 47th number, do you have to work out ALL the first 46 numbers? The answer is NO - thanks to a formula which uses the magic of Phi.

If you want the 47th Fibonacci number, you just put 47 in place of n and work it out. In case you think this is too ugly, here we've replaced the Phi sign with its value:

There are two things to know about this formula...

• You need a really good calculator to work out the powers!
• Even though it's full of square roots of five - amazingly the formula ALWAYS gives a whole number as the answer!

There's more about this strange series on our Fibonacci and Nature page

HOW DO I GET THE FIBONACCI SERIES FROM PASCAL'S TRIANGLE? asked by Rex 231i4k

In above question to find out what the Fib series is.) k5t38

• You need to draw a Pascal's triangle starting with a 1 by itself at the top. (You can find out how to do this on our Pascal's Triangle Page )
• You then draw in a set of diagonal lines like in our diagram.
• If you add up the numbers on the lines you get the Fibonacci series!

It's not too surprising that Pascal's Triangle produces the Fibonacci series in some strange way. After all, each number on the triangle equals the two numbers above it added together and each number in the series are the two numbers before it added together.

WHAT ARE THE FIRST 100 ROWS OF PASCAL'S TRIANGLE? asked by Alice 331u6r

Not enough space here sadly (and exactly WHY do you want to know anyway?) but if you really want to see a Pascal Triangle that sings and dances, go to Dolly's Links and try the The Self-Colouring Pascal's Triangle. It's brilliant! 6j4q4l

HOW MANY DIFFERENT NETS ARE THERE THAT MAKE A CUBE? asked by Bob 5j3dq

In Vicious Circles we find out about "nets" which are flat shapes that fold into solid objects. If you put six squares together you can make a net that folds into a cube. Our challenge was to find out how many ways you can arrange your six squares to start with! If you don't count rotations or reflections, we think there are 11 different nets in total, and you can see them all here in this lovely picture. 1y5hj

Of course you can make other patterns with six squares, but they won't fold together to make a cube!

While we're talking about nets, LOADS of people keep asking us:

WHAT IS THE NET OF A SPHERE? o6q4c

About the closest you can get to it is peel an orange VERY CAREFULLY so that all the skin comes off in one bit then squash it flat. 4s4h17

Pssst... we have to it, that answer makes us feel rather pleased with ourselves!

HOW CAN I FIND THE AREAS OF REGULAR POLYGONS? asked by Steve Charlton 144i6k

If your polygon has n sides and the side length is s then... 3r57x

So if you have a regular hexagon that measures 4cm each side, you use s=4 and n=6. You get:

AREA = 6*4*4 / 4[tan(180/6)] = 24 / [tan(30)] = 24 / 0.577 = 41.569 sq cm.

WHAT IS THE VOLUME OF A TORUS? asked by Shanthan Golden 3h95n

In case you don't know, a TORUS is like a cylinder curled round and ed up to make a circle, which ends up looking like a lifebelt or doughnut shape. 1a406o

There are two formulas for the volume depending on which measurements you've got.

R = the distance from the centre of the hole in the middle to the centre of the cylinder, and r = the radius of the cylinder. 16i3t	a =the distance from the centre of the hole in the middle to the inside edge and b = the distance from the centre of the hole to the outside edge. 1n866

We love these formulas because you get PI squared. It doesn't happen very often!

Why does 0.99999... = 1 ? asked by loads of people! 55566s

You'll find the answer HERE 6m38q

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In the book Do You Feel Lucky? we use a lot of "factorials" in sums. A factorial has a sign like this: ! 324qm

But the BIG question is: what does 0! = ? The answer is also 1. 6h60n

In the book we showed how our first investigations went:

However this wasn't good enough for a supervillain known as "The Mathster". He challenged us to give a proper reason, and this is the best we came up with:

If you have 3 playing cards in your hand, you can arrange them in 3! different ways. In other words you can put them in 3 x 2 x 1 = 6 different orders. (It's all explained in "Do You Feel Lucky?") If you have 2 cards, you can put them in 2! different ways which is 2. If you just have 1 card you can only arrange it 1! ways which is 1. But if you have ZERO cards then there's also one way of doing that! 6z52b

You might have to think about the last bit like this: if there was NO way of having zero cards, then wherever you went or whatever you were doing, it would be impossible not to have at least one card on you. Even if you were in the bath. 326242

So 0! = 1. 1u4270

That was our best answer, but several other Murderous Maths fans sent in their answers. We especially liked this one:

GAIL WEISS says: 4! = 4x3x2x1 and 3!=3x2x1. Therefore 4!=4x3! In the same way 3!=3x2! and 2!=2x1! So it follows that 1!=1x0! Therefore 0! must =1 or 1! would be 0 and so 2! would be zero and then 3! would be zero...and so on.

Let us hope that The Mathster now stops bothering us with tough questions and blends back into his secret civilian identity which is being a Murderous Maths fan called David Small.

WHAT DOES 0/0 EQUAL? asked by Alex Jeffries 4k2e72

Alex went on to ponder: Does 0/0 = 1 because both numbers are the same, or is it 0 because there's nothing involved? 3d5v40

We had fun checking this one, but our final results are something like this:

If you have a bag containing 0 chips and share them between no people, how many do they each get?

However somebody called "KOO C W" came up with this for us: 1w473f

Because anything x 0 = 0 therefore 0/0 can = anything.

Good one!

HOW CAN YOU FIND THE VOLUMES OF THE REGULAR SOLIDS? asked by Joe Symons-Smyth 646733

As Joe pointed out, you can easily get the volume of a cube by measuring one side and cubing it. So if the side measures "s" the volume of a cube = s3. 1p6q36

The challenge was to find formulas for the other regular solids!

We've finally got the answers which are on their own special page: Regular Solid Volumes.

WHAT IS THE HIGHEST PRIME NUMBER EVER CALCULATED? asked by "MooCow" 5w6x3r

Good question! You can get the answer yourself on one of our favourite websites: 4t4c4i

News of the BIGGEST prime number found so far!

WHAT IS THE DIFFERENCE BETWEEN A NORMAL PRIME AND A MERSENNE PRIME NUMBER? asked by Piyush "Fat Boy" Singh 2r2l73

In the NUMBERS book we meet Mersenne primes which have to fit the formula (2n-1). By the way, not all numbers that fit the formula are primes, just some of them. 5s6am

So a NORMAL prime can be any prime number, but to be a Mersenne prime when you add 1 it must make a power of 2.

• 17 is prime but it is NOT a Mersenne prime because 17+1 = 18 which isn't a power of 2.
• 31 IS a Mersenne prime because 31+1=32= 2⁵

EXACTLY WHAT IS AN OBLONG? We were told this by Katherine Barker 5a5z5m

In the dictionary, "oblong" means that a shape is longer than it is wide, so if you have a shape called an oblong it could be an ellipse, but more usually it's a rectangle. It's the rectangle bit that's confusing so just to clear things up: An oblong can be a rectangle but it cannot be a square. A square is a rectangle, but it is not an oblong. A rectangle can be a square or an oblong. 1f1p4i

HOW CAN I WORK OUT THE AREA OF A TRAPEZIUM IF I ONLY KNOW THE LENGTHS OF THE FOUR SIDES? asked by "Roberdin" 470r

This is a REAL beauty! Normally to work out the area of a trapezium you need to know the height between the parallel sides. In this diagram the area would be: 68185b

But what if you don't know the height or any angles? At first we thought it was impossible, but lots of people told us differently!

Here's the formula we finally worked out for the area of a trapezium where sides a and c are parallel:

As you'll be desperate to know how we got this amazing result, take a look at our special TRAPEZIUM AREA page!

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