Mathmatters

The Hidden Calculations of Everyday Life

Chris Waring

• Éditeur: Michael O'Mara
• Année: 2022
• ISBN:
  9781789293685
• Sujet: Matematica
• Sous-sujet: Aritmetica

Mathmatters is a humorous guide to the hidden calculations that are essential to everything we do. From making a cup of coffee to negotiating traffic to selecting candidates for an interview, we can't make it through the day without employing some essential mathematics. Did you know that there are some serious calculations involved in making the perfect cup of coffee (involving ratios)? That an understanding of Braess's paradox will mean you can remain calm about road closures on your commute as they may make your journey faster (using equations relating to speed/distance/time)? Or that your online shopping habit can teach you about game theory (mathematical models of strategies)? Full of easy-to-understand mathematics and fun, if not entirely helpful, illustrations, Mathmatters is your essential guide to understanding the rules and measures that surround us every day, and determine the outcome of every move we make, every button we press and much of our decision-making, whether we are aware of it or not.

PART 1

RISE AND
SHINE

From the moment you wake up, mathematics is part of your day.

In the next few chapters we’ll take a look at the governing principles behind your morning cup of coffee, hitting the gym and why you sound so damn good in the shower.

CHAPTER 1

Wake Up and Smell the Coffee

I don’t know about you, but I do not consider myself fully functional – or even partially functional, to be honest – until I’ve had my morning coffee. I silence the spiteful bleating of the alarm and stagger downstairs, barely conscious – barely even a vertebrate. I get out the cafetiere and all the other necessaries while I boil some water. Once that first sip hits my system, the magic begins.

I am not alone. Around 35 per cent of the world’s population have a coffee every day, with Scandinavians drinking the most coffee per head. Sixty per cent of adults in the US drink coffee. Four billion cups of the miracle fluid are drunk every year worldwide and the industry is worth over £100 billion. Coffee provides work for over 100 million people, from farmers growing beans to baristas brewing shots.

Coffee became a major thing in North Africa and the Arabian Peninsula in the late fifteenth century, when people invented a drink made from the beans of a bush native to Ethiopia that the local goats and birds loved to chew. The wonderful properties of the drink meant its popularity spread, arriving in Europe in the 1600s with emigrants from the Middle East setting up coffee houses in many cities. From here, cafe culture caught on and coffee trees were planted all over the world.

Previously, Europeans tended to drink alcoholic beverages. The brewing process made the water free of unpleasant diseases such as typhoid and cholera, but the alcohol had serious intellect-dampening side effects. Coffee, on the other hand, improves memory and focus, and so coffee houses became meeting places for intellectuals of every type, independent of class and wealth. Scientists, economists, politicians and revolutionaries would all hold forth in the coffee houses and anyone could join in. Some historians believe that the Enlightenment – the name given to a blossoming of intellectual movements of the seventeenth and eighteenth centuries – started in these coffee houses.

How Much?

The mathematics of coffee brewing is extremely complicated. Modelling thick, lumpy fluids such as coffee – and, indeed, blood and soup – and how they move around is an ongoing mathematical endeavour. We can look at simpler concepts though, such as how much coffee should I put in the cafetiere in the first place? Given my medium-strength ground coffee, I can alter the strength of my drink by altering the ratio of coffee to water. Typical ratios, by mass, vary from 1:10 for a mind-bendingly strong cup of coffee to 1:16 for a paler, less palpitation-inducing brew. I would need to drink more of the weaker coffee to get the same amount of caffeine, and obviously the less water I use, the more intense the flavour of the coffee will be.

How do ratios work? I have discovered through experimentation (I’m nerdy like that) that one gram of coffee for every thirteen of water works well for me. I can show this with the ratio 1:13. Ratios are good because they don’t rely on any particular units, provided you take the same type of measurement (for example, mass with mass, volume with volume, etc). The ratio will produce the same strength coffee whether I use one gram of coffee grounds and thirteen grams of water or one ounce of coffee grounds and thirteen ounces of water. Provided that I use the same unit for both the coffee and the water, I’ll get the strength of coffee I like, whether I measure it in tonnes, elephants or Chinese liăng (which is equivalent to 50 g, in case you didn’t know).

I don’t want to be fiddling around with a gram of coffee at a time while I’m standing in my pyjamas on a cold winter’s morning, though. Another beautiful thing about ratios is that they are easy to scale up. When I multiply each side of the ratio by the same amount, I’ll get different numbers but the same relative proportions. For instance, if I multiply each side of my ideal 1:13 ratio by 20, the ratio becomes 20:260. This tells me I should weigh out 20 g of coffee grounds and pour on 260 g of water. We tend to talk about amounts of water in terms of volume rather than mass – the size of the water rather than how heavy it is – but water conveniently has a volume of 1 millilitre for every gram, so I need to put 260 ml of water in the kettle. This means I’ll get about 260 ml of coffee when I pour, which is enough for a good cup of coffee with room for a little milk.

It’s still a bit of a faff to weigh out 20 g of coffee every morning, especially when I can barely keep my eyes open. It’s much easier to scoop spoonfuls of coffee grounds into the coffee pot. A heaped dessertspoonful of my coffee grounds weighs about 12 g. How much water does this require? Well, we can see from the original ratio of 1:13 that I need thirteen times as much water as coffee grounds. So, 12 g of coffee will require 12 × 13 = 156 g of water, giving a ratio of 12:156. This clearly won’t give as much coffee as the 20 g. If I use two spoonfuls, I double the amounts, giving 24:312. This gives me coffee of the required strength and enough for a bit of a top-up. Great – I can jumpstart my day without the bother of getting the scales out every morning. Of course, I still have to be careful to boil and pour out the right amount of water, but using the right size of cafetiere for your tastes helps with judging this.

Upping Your Coffee Game

I’m sure you know someone who takes their coffee very seriously. Maybe a little too seriously. They may have an espresso machine, and use artisan-ground coffee that’s been through a civet’s colon; that kind of thing. Mathematically, if I became more serious about coffee, I could introduce other variables to my simple coffee ratio. I could buy beans instead of ground coffee and vary the fineness of the coffee grounds. I could also alter the length of time that the coffee brews for, or change the temperature of the water.

Coffee is chemically very complex and contains over 1,800 ingredients. As the hot water interacts with the coffee grounds, some of those ingredients dissolve faster than others. If you don’t brew the coffee for long enough, you end up with more of the fast-dissolving sour flavours, as well as a weaker brew. Leave it to brew for too long and your drink can be overwhelmed by the slow-dissolving bitter-tasting chemicals. Somewhere in the middle is the Goldilocks zone, where sour and bitter combine to make the smooth, rich, caramel flavour that is the hallmark of a good coffee.

The size of the grind affects this, too. The smaller the grind, the faster the coffee will brew compared to a larger grind. Mathematicians at the University of Limerick have been researching the science of making a consistently great cup of coffee. They have modelled the entire process as a system of equations, taking into account the type of roast and the chemistry of the water used, as well as the method of brewing. This complicated mathematical model allowed them to test different combinations without having to physically brew the coffee.

They also made an important discovery to do with the grind size. Many cafes grind their coffee very finely. This works well in espresso machines, where very hot (but not quite boiling) water is forced through the coffee grounds at high pressure to make a very intense ‘shot’ of coffee. The model showed, however, that if the coffee is too finely ground, the granules clump together and behave like much larger granules, which reduces how much coffee can dissolve after all. The solution is, therefore, to use a slightly larger grind size, which also reduces the amount of coffee required. Customers get a better cup of coffee, cafes use fewer coffee beans, and the environmental impact of the coffee industry is reduced.

Breakfast of Champions

Coffee makes you feel like a superhero, but how much coffee would you need to fuel actual superhero powers?

Even without dipping into the pages of a comic book, caffeine is a stimulant that acts on your central nervous system to produce several effects. The stimulation puts your body into fight or flight mode, so adrenaline is released into your bloodstream. This hormone increases your heart rate and blood pressure, expands the air spaces in your lungs, dilates your pupils to let in more light and diverts blood to your major muscle groups.

Exactly like a superhero.

There’s more, though. The caffeine also interferes with your brain’s interaction with another hormone called adenosine, which is crucial for alertness. Adenosine accumulates in your brain while you are awake. The more adenosine detected by your brain, the sleepier you feel. Caffeine stops your brain from detecting the adenosine, making you feel more awake, alert and ready to do tricky maths problems.

So, coffee will make me super-fast and super-clever, with super reactions and super strength, but what if I want some proper superpowers, like being able to shoot lasers out of my eyes?

A normal torch emits light of different wavelengths (which we interpret as colour) that fan out from the torch in a wide beam. In a laser, the light is all of the same wavelength and travels in the same direction, giving a nice tight spot that will drive your cat – literally and figuratively – up the wall. Light is a form of energy, so a powerful laser can be used to burn or cut things. Laser eye surgery uses this idea to make very fine cuts into your eyeballs. But I want lasers to come out of my eyes.

Imagine the maths hero QED has to rescue his sidekick Square Root from the evil clutches of their arch-nemesis The Guesser. Square Root is trapped in a steel box suspended over the lava pool in The Guesser’s volcanic headquarters. Only QED’s laser eye vision can save the day!

Lasers are rated in terms of their power...