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Maths In The Head

Maths In The Head
It doesn’t hurt, however, to know a few of the following tricks to make ball-parking easier and maintain the slightly dubious reputation engineers have for “doing maths.”

It turns out there is one thing that everyone knows about engineers: We Do Maths. I’ve lost count of the amount of times friends have simply deferred any arithmetic to my good-self because “you’re an engineer- you do maths“; dividing bills, working out petrol costs and calculating quantities- my apparent skills have no end. It’s got to the point where my somewhat-taller-half no longer actually does any maths, he just wonders out loud any arithmetic he requires the answer to and then stares meaningfully at me until I answer.

The slightly worrying thing about this whole situation is that I’ve also lost count of the number of times I’ve witnessed an engineer (myself included) struggle with a trivial calculation, joking: “it’s lucky I don’t do maths for a living…”. Of course, the fact is that the sort of maths engineers use on a daily basis has a habit of being a little more abstract and calculator based than your day-to-day restaurant disagreement. It doesn’t hurt, however, to know a few of the following tricks to make ball-parking easier and maintain the slightly dubious reputation engineers have for “doing maths”.

Double and Halve

People are good with the number two; whether it’s because it’s the first times-table most of us learn, or just because doubling and halving quantities comes naturally to us. By appreciating that we can almost instinctively divide and multiply by two you can substantially improve your mental arithmetic speed; it’s called the double-and-halve trick. When multiplying two numbers just halve the smallest and double the largest until you end up closer to a factor you’re familiar with:

793 \times 24 = 1586 \times 12 = 3172 \times 6 = 6344 \times 3 = 19032

Obviously this trick is at its most effective when you have a number that is a power of two. For those times when life gives you lemons, however, you can still achieve similar speeds by separating out the difficult portion:

123 \times 13 = 123 \times (12 + 1) = 123 \times 12 + 123 = 246 \times 6 + 123 = 492 \times 3 + 123 = 1476 + 123 = 1599

Form the Square

Squaring numbers is a common thing in engineering; probably because of they way we work with physical quantities such as area. This trick is actually the application of a simple rule, and an equation that everyone learns and immediately forgets. For those of you who don’t know it, there is a rule for squaring any number ending in five: form a number using all but the last digit (i.e. 5), multiply that by itself plus one, and then finish it with 25:

35^2 = ( 3 \times 4 ) \; \text{and} \; ( 25 ) = ( 12 ) \; \text{and} \; ( 25 ) = 1225

It even works with decimals, just temporarily remove the point by multiplying though tens:

1.05^2 = \left(\frac{10.5}{10}\right)^2 = \frac{ ( 10 \times 11 ) }{ 100 } \; \text{and} \; ( 25 ) = \frac{110}{100} \; \text{and} \; 25 = 1.1025

Of course, you’d be lucky if you managed to pull off a calculation where you only had to square something ending in a five. However, you are only ever ±5 away from a number ending in 5, and that’s where our GCSE maths equation makes an appearance:

(x + y)^2 = x^2 + 2xy + y^2

Using this little gem, we can split our non-conforming number into something ending in five, and a number between -5 and 5 that we should be able to square in our heads. This allows us to use the ‘ending-in-five’ trick even when the number doesn’t:

39^2 = (35 + 4)^2 = 35^2 + 2 \times 35 \times 4 + 4^2 = (3 \times 4) \; \text{and} \; (25) + 2 \times 70 \times 2 + 16 = (12) \; \text{and} \; (25) + 280 + 16 = 1225 + 296 = 1521

Or, similarly, where you nearest five is greater than the number you’re trying to square:

44^2 = (45 - 1)^2 = 45^2 + 2 \times 45 \times (-1) + (-1) ^2 = (4 \times 5) \; \text{and} \; (25) - 90 + 1 = 2025 - 90 + 1 = 1936

Engineers Aren’t Mathematicians

This final trick isn’t really a trick at all, it is just the recognition that engineers aren’t mathematicians; we operate in uncertainties and precisions. Whilst I can’t calculate the square root of 73 in my head, I know that the answer is between 8 and 9, and is closer to 9; at a guess I’d say [Ed. you’ll have to trust me on this one:] 8.6, which (at 73.96) is good enough the majority of Civil Engineering applications.

By keeping an eye on the degree of accuracy you’re simplifying each stage by, you should be able ensure that you can calculate to a good enough aim for the application you need. You can improve your accuracy by acknowledging that, as an engineer, you’re trained to think conservatively and therefore you will want to normalise your conservatisms by taking geometric means between ranges (see Fermi Estimates for a guide).

It should be noted that being ‘close-enough’ when dealing in bistromathics is, alas, never good enough!

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