Live interview with BBC Radio Cambridgeshire on how maths can help you to make decisions in life… plus a few examples of my own (good and bad).
Live interview on BBC News about the legacy of Iranian Mathematician Maryam Mirzakhani who tragically passed away today (July 15th 2017). She was the first female winner of the Field’s Medal – the mathematical equivalent of the Nobel Prize.
I strip-back the most important equations in maths layer by layer so that everyone can understand them. This time it’s the turn of the most beautiful equation in maths…
Question number 3 sent in and voted for by YOU… enjoy!
The mascot of the Pokémon world and probably the most famous of all the Pokémon: Pikachu. It’s pretty much just a short, chubby rodent with two red circles on its cheeks that it apparently uses to store electricity. So the obvious question to ask here then is how many Pikachus would it take to power a lightbulb?
The official Pokédex tells us that Pikachu is able to ‘release electric discharges of varying intensity’ and is also known to ‘build up energy in its glands that needs to be discharged to avoid complications’. Quite what these ‘complications’ might be I’m not sure – suggestions on a postcard. The Pokédex also says that the tail of a Pikachu plays an important role as it acts as a grounding rod to prevent the creature from electrifying itself and also allows one Pikachu to recharge another one that’s running low on juice. Another fun fact about Pikachu’s tail is that a female will have a v-shaped notch at the end which is not present on males. Next time you’re out Pikachu hunting, now you know.
When a Pikachu uses its signature move, thunderbolt, it releases 100,000 volts of electricity. You might think that this would be enough to kill you, never mind just power a lightbulb and you wouldn’t be wrong, but there’s a little more to it than that. When someone dies from electrocution, it isn’t the voltage that kills them, it’s the current. If a current of just 7 milliamps reaches your heart for around 3 seconds its lights out. The current is tiny, but it disrupts the hearts natural rhythm causing it to stop. You can of course be electrocuted and live to tell the tale, but this means that you were very lucky in that the path that the electricity chose to take through your body must have avoided your heart. In general, electricity will flow along the path of least resistance, as with most things in nature it’s lazy and so takes the path that is the easiest.
I’ve now mentioned three terms, which means a good old fashioned physics formula triangle. The particular one we want here is Ohm’s Law:
The resistance of your body is around 100,000 ohms, which means thunderbolt generates a current of 1 amp that will flow through your body. In other words, if it hits your heart you’re toast.
Now onto lightbulbs. They come in all sorts of different shapes and sizes, and different amounts of wattage or power. The standard one is 60 watts and that’s the one we’re considering here. None of these fancy energy saving bulbs that start off really dim so that you can’t see a thing and gradually build up in brightness… (don’t get me started on them). The key bit of physics that we need relates power (in watts) to Ohm’s Law above and states:
For a 60 watt light bulb attached to the UK mains supply of 230 volts, this generates a current of 0.26 amps. If we were to connect the bulb up to a Pikachu using thunderbolt the current decreases hugely to only 0.0006 amps. The light bulb will still work but the current will just be very small. This means only a small wire is required and also has the added benefit of reducing the chance of electrocution… However, 0.0006 amps is still enough to kill you if it hits your heart, so be careful next time when you’re hooking up your lightbulb to a Pikachu!
My first ever live radio interview from almost 2 years ago – enjoy! You can listen here.
The Millennium Prize Problems are a set of 7 maths problems that have been deemed so important that if you can solve any of them, you’ll be awarded 1 million Dollars. Graihagh Jackson spoke to Cambridge University’s Tom Crawford to find out what these problems are…
Tom – So, yeah. These are sort of a second reincarnation of the idea of important maths problems. So, in the year 1900, there was a mathematician called Hilbert who actually came up with 23 problems that he considered to be the most important in the year 1900. In fact, one of these is actually one of the millennium problems as well. And then the year 2000 came around, it’s a big event, lots of things happened, and maths thought, “Let’s get involved with this.”
Graihagh – Jump on the bandwagon.
Tom – Exactly. They sort of sat down and said, “Right. These are the biggest unsolved problems in mathematics at this time.
Graihagh – Did they say 7 because 7 is a prime number and an interesting maths number?
Tom – I’m not sure exactly why they chose 7, but that sounds like a good idea to me.
Graihagh – And you say ‘they’, these mathematicians, who are they and I assume, they’re the ones fitting the bill of a million dollars, right?
Tom – Yes. So, this is the Clay Institute. It’s an institute based in America. They had money left to them and thought, “Let’s use this as prize money as a sort of extra motivation.” But if you ask most mathematicians – I won’t say all – if you ask most mathematicians, they wouldn’t actually try and tackle these problems for the money. It’s more just for the fun of doing the maths in some sense.
Graihagh – So, can you give me a quick rundown of the final 7.
Tom – I’ll start with the one I’m doing my PhD on, this is the Navier-Stokes equations. This is fluid dynamics. This is a set of equations that model the flow of every fluid – so water, air, anything. Then we’ve got the Mass Gap Hypothesis. You can think of this one as asking the question “why do things have mass?”
Graihagh – Sounds interesting. So, you’ve mentioned 2 of them there. There’s 4 more by my count, by my math.
Tom – 5
Graihagh – 5? Oh no! Okay, what are the other 5?
Tom – We’ve got the Poincare Conjecture which is an interesting one as we’ll find out later. We’ve got the Riemann Hypothesis which looks at prime numbers. We have P versus NP – one of the more famous ones. So, this is basically looking at how computers work. And then the last two are quite abstract. So, you have the Birch and Swinnerton-Dyer Conjecture.
Graihagh – Wow! Great names.
Tom – And then finally, the Hodge Conjecture. So, the Hodge Conjecture is quite an interesting one because depending on which expert in the field you ask about this problem, they will give you a different definition of what this problem is.
Graihagh – So, no one can agree what the actual problem is in the first place?
Tom – Yeah. That’s how complex some of these problems are.
Graihagh – Wow! So, how are we ever going to go about solving them if we can’t even agree what they are in the first place?
Tom – Well, I mean some of them are a bit more approachable and there’s a bit more general consensus about what this problem is. But I think the main idea of these problems it’s not necessarily getting a solution, it’s understanding more. So, by trying to conquer this giant mountain of a problem, you’ll scale smaller peaks along the way and make new advances in other areas of maths.
Graihagh – A million pounds sounds like an awful lot, but I suppose if you spend your entire life doing this then actually, it’s not that much.
Tom – No. Sort of estimates of the amount of time it would take to solve one of these problems, it actually works out as being paid below minimum wage for the amount of time you would need to spend to solve one of these problems.
Graihagh – Wow! We’ve got some very dedicated mathematicians out there.
Tom – Yeah, most mathematicians I think actually do it for fun, if you can believe that.
Graihagh – I’m just thinking about my maths class back when I was 16 and maths certainly wasn’t my favourite subject but maybe I should revisit that.
Tom – Yes, sounds good.
7 – SEVEN
‘I’ve sailed the seven seas, yarggg’ says a drunken pirate… There are of course more than seven seas – see how many you can name – but the point is that by seven we have reached what would have been considered a pretty large number in the past, especially amongst the less well-educated, such as pirates. We do, however, have seven days of the week. The reason for this is thought to be due to the Babylonians, who measured time using the sun and the moon. The sun appears once each day and the moon once about every 29 days, making it pretty much once a month. The use of the seven-day week was probably because they wanted a smaller measurement between a day and a month and the best you can do with 29 is 4 lots of 7, with one left over which I guess they just ignored.
The Romans are behind many of the names we use today for the days of the week, though the English ones have also been influenced by the Angles and Saxons, back when the Vikings were in charge. Monday is named after the moon, Sunday the Sun, Tuesday after Tiw – the Norse God of war, Wednesday after Woden – the chief Norse God, Thursday after Thor – the guy with the hammer and the God of thunder, Friday after Frigga – the god of marriage and finally Saturday after Saturn – the Roman God of time and harvest.
Seven is also the maximum number of circular objects that can be securely tied together in a bundle. It’s a lovely geometrical problem using the idea that exactly 6 circular objects will fit around a central circular object without leaving any gaps – try it out now with 7 coins on a table. If you try to add any more than 6 around the central one gaps will appear, which would cause your bundle to fall apart.
8 – EIGHT
The number of legs on a spider, the number of tentacles on an octopus and now the number of planets in our solar system. There of course used to be nine until Pluto was demoted to ‘dwarf planet’ back in 2006. The reason? Astronomers found another dwarf planet called Eris that has a larger mass than Pluto despite being some three times further out, which ultimately led to the reclassification of what is and isn’t a planet.
Eight is a big deal in Asia, like a really big deal. Its pronunciation in Chinese sounds the same as the word for prosperity and so its deemed to be a good luck charm. It also plays a big role in Chinese philosophy with the eight compass points – North, North East, East, South East, South, South West, West, and North West – each being assigned to one of eight moods and personalities, eight natural features and eight members of a family: mother, father and three children of each sex… no wonder they have an over-population problem.
9 – NINE
The last single digit number and one that we seem to like a lot as humans, maybe it’s something to do with the 9 months we spend inside our mother’s womb before birth… It’s popular in the animal kingdom too – cats are supposed to have 9 lives (unless you’re Spanish in which case they have 7).
You can also do a lot of fun mathematical tricks with the number nine. Take any number and multiply it by 9, then add up all of the digits, what do you get? Let’s try it out. 877 x 9 = 7893 and 7 + 8 + 9 + 3 = 27 and 2 + 7 = 9. This will happen every time. There are others too: take any three-digit number, switch the first and last numbers around, and then subtract the smaller one from the bigger one. The middle digit of the number left is always a 9! That second one is a great way to trick your friends into thinking that you can read their mind…
The reason it works can be worked out using algebra. It’s a word that has a bad reputation but don’t let it put you off because it isn’t really that scary… (not until you get into n-dimensional spaces anyway). Let’s call your three-digit number a b c, where each letter represents the digit. First thing we do is switch the first and last ones around so now we have c b a. Before we can subtract the two numbers we have to make sure we keep track of which digit is in the tens column and which is in the hundreds column. So for our first number, we have 100a + 10b + c and for our second one we have 100c + 10b + a. Now subtracting one from the other we get: 100 (a-c) + (c-a). So importantly the tens column is empty, but remembering that we subtract the small number from the big number means that (a-c) is positive and so (c-a) is negative. That means we have some number of hundreds, minus a single digit number. The smallest we can be left with is 91 and importantly there is no way to get anything other than a 9 in the tens column. Boom. There’s no such thing as mind-reading – sorry Derren – and we’ve just proved it using maths.
My thesis is based on experiments. A weird thing for a mathematician to say you might think, but that’s the truth. It was always planned to be experimental in nature – it even says so in the title – and that’s because it isn’t practical to go to the nearest large scale river outflow (for my work that would be the Rhine in the Netherlands) and start trying to measure things. Fieldwork works well on a Geography trip: you put your wellies on and start splashing around in a stream, measuring the depth with a metre ruler and the speed of the stream by timing a little paper boat as it sails downstream… But things aren’t so easy when you’re talking about a river several kilometres wide and tens of metres deep. The bigger rivers are harder to measure, but the big rivers are precisely the ones that we need to look at, because they’re the only ones big enough to be affected by the earth’s rotation. But we’ll come to that. First up, how do we recreate a big river in the lab?
The trick is to scale things down, as you may have guessed, but there’s a little more to it than just building a scale model of a river. Those little wooden models of a city or building that architects use to help see how their plans will come to life are scale models of the real thing: they are built the same, just with all measurements at a ratio of 1:500 say of the real distances. For example, if a real-life football pitch is 100m in length and 75m wide, then for a 1:500 scale model it would be 20cm long and 15cm wide. A scale model is a good idea in principal, but when working with rivers that are 1km wide and 10m deep, once you scale it down to lab-size, your depth is about as thin as a piece of paper, which isn’t practical. We have to be a little cleverer as mathematicians and think about what properties of the river are the most important and then only include those in our lab model.
I’ll give you an example. Let’s suppose we are trying to work out how fast Usain Bolt travels when he runs the 100m. For ease of the numbers, we can say he runs 100m in 10.0 seconds (a slow day for Usain – he’d had a few too many chicken nuggets before the race). There are many other factors that will affect his speed:
- The wind was blowing at a speed of 1m/s against him
- It was raining
- He was wearing a waterproof coat (he forgot to remove it)
- His bodyweight was 2kg higher than normal (those damn chicken nuggets)
- One of his shoes was missing a spike
- The race was in Brazil
We know that all of these things will affect Usain’s speed, but which do we actually think are important enough to include them in our model? If we ignore them all then at a first guess, we can just use the speed = distance/time triangle from school which gives 100/10 = 10m/s. This would be a first order estimate using just two properties: time and distance. If we want to include more information and get a more accurate answer, then maybe we can include the wind speed: 1m/s against him means he must run at 11m/s to cover 100m in 10 seconds. This is an increase of 10% on our first estimate of his speed, and so probably quite important. Because of the direction of the wind the rain will act against him too, though probably by only a very small amount. The coat will increase the air resistance slowing him down, but again probably quite small.
The key point here is that there are many factors that will affect the speed of Usain Bolt as he’s running the 100m, but as mathematicians our job is to figure out which ones are the most important and to only consider those. If we tried to model every small effect things would get very complicated very quickly and we don’t want that (trust me I’ve tried). Simple is good – so long as you don’t ignore the important bits…
For Usain’s speed we can probably keep the distance, time and the wind speed and that’s about it. Even if we included everything else I doubt the value would change very much from 11m/s, certainly by less than 10% which is a nice acceptable error that we can live with. For scaling rivers down to work with them in the lab we have to do the same thing: pick the important parts of the problem and ignore the rest. The key thing is picking the right bits – which we’ll come onto next time.
You can read the first article about my thesis here.
A new series where I strip-back the most important equations in maths layer by layer so that everyone can understand them… First up is the Navier-Stokes equation.