Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything. An entertaining introduction to mental maths and problem solving

by Rob Eastaway

‘Another terrific book by Rob Eastaway’ SIMON SINGH ‘A delightfully accessible guide to how to play with numbers’ HANNAH FRY How many cats are there in the world? What's the chance of winning the lottery twice? And just how long does it take to count to a million? Learn how to tackle tricky maths problems with nothing but the back of an envelope, a pencil and some good old-fashioned brain power. Join Rob Eastaway as he takes an entertaining look at how to figure without a calculator. Packed with amusing anecdotes, quizzes, and handy calculation tips for every situation, Maths on the Back of an Envelope is an invaluable introduction to the art of estimation, and a welcome reminder that sometimes our own brain is the best tool we have to deal with numbers.

Genres: Nonfiction, Science, Mathematics, Stem, Education, Reference, Popular Science

208 pages, Paperback

Published June 15, 2021

About the author

Robert Eastaway is an author who is active in the popularisation of mathematics. He is a former pupil of The King's School, Chester, England and has a degree in Engineering and Management Science from the University of Cambridge. He was President of the UK Mathematical Association for 2007/2008. Eastaway is a keen cricket player and was one of the originators of the International Rankings of Cricketers. He is also a former puzzle-writer for the New Scientist magazine and he has been involved in live mathematics shows on Virgin Radio (now Absolute Radio). He is the Director of Maths Inspiration, a national programme of maths lectures for teenagers which involves some of the UK’s leading maths speakers such as Simon Singh and Colin Wright.

Robert Eastaway's titles appear under the name "Rob Eastaway" in the UK and some other countries.

Reviews

[Domhnall · Rating 5 out of 5]

Calculators, computers, Google, Wiki, we can leave maths to machines, but we are left with the problem of interpreting numbers when we encounter them in everyday life, from special offers to politicians' promises. Is a billion pounds extra spending on the NHS a lot of money or loose change? How many marchers joined a protest - and do we believe the police or the organizers based on faith or based on some reasoning? In a lot of situations we need a reasonable number, we probably want it reasonably quickly, we do not need it to be accurate, and we need to be suspicious of numbers that claim to be accurate.

This is a fun book, not over lengthy or challenging, but it does require some effort (really not a lot - back-of-an-envelope stuff) from the reader. We can calculate in our head on or the back of the proverbial envelope, but we still need to rehearse our elementary numeracy skills, put in some effort and use some imagination. That said, it is very rewarding, in its way, and might make a difference in practical, real world situations.

[John · Rating 3 out of 5]

The book is split into four parts, one about precision, one exploring the essential tools of estimation, one with estimation examples and one with curious "Fermi" problems.

By far the most useful of these is the part which deals with equipping the reader with tools and tricks for calculating or estimating answers to mathematical problems. There are some really useful techniques here, e.g. 16% of 25 sounds hard to do in your head but when you know it's the same as 25% of 16 it becomes trivial. The author introduces his own Zequals estimation technique for a cavalier approach to problem solving but then only spends a few pages on it.

A lot of the book was dedicated to examples of using estimation to come up with reasonable answers to obscure questions, e.g. how many cats there are in the world but sometimes seemed to use numbers no sane person would know! The Fermi chapter I found less interesting.

Even though I knew some of them before I will certainly remember some of the techniques from the essential tools chapter I haven't used in practice. It's just a shame the book didn't delve a bit deeper into these and other techniques.

[Adelyne · Rating 5 out of 5]

This was the fun read that I expected from the title, and on such an important topic too - though this book really drives home the point that it really is all about (increasingly uncommon) common sense. I learnt a trick or two in this one, which was quite nice, despite the fact that I think I am relatively quick at mental arithmetic and have developed tricks of my own previously.

Eastaway writes well, even in the earlier sections that are a bit more maths-ey I didn't find dry at all, with what I thought was just the right level of humour and conversationalism in the style. I read this in 5 sittings (1 for each of the first 3 chapters, and then broke the last one down in two as I was attempting the problems and it was starting to do my brain in), and especially for the first two chapters it worked very well as his tips and tricks do chain into one another.

The last section on Fermi problems I particularly enjoyed, it was fun to (try and) work through them - even though I didn't manage them all - and then to read about how the author would have approached the problem. I see from the reviews that the author has other books, and I will be checking them out. 5 stars and onto the keepers shelf.

[Maurizio Codogno · Rating 5 out of 5]

Spannometria pratica

Rob Eastaway è un noto divulgatore matematico britannico. In questo libro ) si è dedicato alla spannometria. In realtà il suo approccio è più ampio. La spannometria vera e propria - i problemi di Fermi - sono solo verso la fine, mentre nelle sezioni iniziali del libro Eastaway mette in guardia contro la troppa precisione e insegna un po' di trucchi del mestiere, oltre che introdurre il suo Zequals (un uguale disegnato coi dentini a sega, per dire che si taglia via tutto tranne la cifra più significativa). Direi che il libro è assolutamente valido per tutti, giovani e meno giovani; leggendolo, imparerete anche a saper non usare la calcolatrice, o almeno a saperla usare con cognizione di causa!

[Jan de Vries · Rating 4 out of 5]

Dit boek is eigenlijk wel een beetje wat ik had verwacht. Het staat vol met tips en handige methodes om snel te kunnen rekenen. Het is vrij makkelijk om alles te volgen en is niet echt van hoog niveau. Dat is op zich niet erg. Het boek bevat goede voorbeelden van hoe redenaties verkeerd gaan en de tips die het bevat zijn erg waardevol. Het is best handig om deze op een plek te hebben.

Dit boek is voor mij als student bij de Natuurkunde docent opleiding erg handig. De tips die er in staan zijn erg bruikbaar bij het snel doen van rekenvoorbeelden op het bord of om mee te geven aan leerlingen.

Er staat verder niet heel veel bijzonders in het boek. Tips, voorbeelden en oefeningen om te maken. Achterin het boek staan de antwoorden en vele leuke uitbreidingen op wat er eerder in het boek aan bod kwam. Ik vond het vooral leuk dat er achter bij stond hoe andere mensen een bepaald probleem in hun hoofd aanpakten. Dat is interessant om rekening mee te houden als docent.

[Rynne Leadbetter · Rating 5 out of 5]

I loved reading this book. I'm really interested in maths and I think it's a really good book for the person sitting in maths class asking "when are we ever going to use this?" My favourite non fiction book that I own hands down.

[Robert · Rating 4 out of 5]

Rob Eastaway has authored a clever little book with lots of practical ways for performing estimations off the cuff (or on the back of an envelope as the title describes it). I really enjoyed this book with handy tools for making quick estimations and good anecdotal stories to show some uses, explain misconceptions and give the reader a feel of how to use some of the practices explained in the book in real life situations. This book is not just for people that are more mathematically minded but is equally if not even more useful for people who consider themselves mathematically challenged, with practical tips even for performing quick estimates for shopping bills to ensure you are not being overcharged to estimation of large distances, unit conversion estimations, odds for winning lotteries etc. While reading this book you quickly realize that none of his methods are beyond people even with the most rudimentary school mathematics. This book is wonderful, I highly recommend it.

[Norm Tresadern · Rating 4 out of 5]

Loved this book! Picked up some really useful tips and it actually felt as if my brain was getting switched on. (If you don't have an envelope handy, napkins work too!)

One star dropped because there was a section at the end that was less interesting to me personally. It was about estimating, and assumed you already know or can estimate some things - like how long it takes to fly from London to Sydney, or that you can visualise a race track and, from that, estimate the width of a lane. I have no idea, and my brain thinks it's trivia, and refuses to remember it. If you already know some or all of this stuff, or if it appeals to you, then Rob shows you how you can use this info to make rough estimates about other things. It was still interesting to read though, even though I'd put my envelope down at this point.

Plus I can now quote pi to 8 significant digits, which I couldn't before, so that's good! 😁

[Ville · Rating 3 out of 5]

Teaches you to, for example, quickly estimate the square root of large numbers, such as 52957081. (That's something in the order of 7000 and a bit.) Sometimes feels like a primary/secondary school math book, but when you get over those parts, it has some neat tricks and even anecdotes about Who Wants to be a Millionaire. 😎🤓👍

[Jez · Rating 4 out of 5]

Very interesting. Liked the tips, wish there were a few more rather than so many examples

[Andy · Rating 5 out of 5]

Interesting!

Really good read and practice at mental arithmetic. No maths degree required, but it helps if you can remember your times tables!

[Matt Cannon · Rating 4 out of 5]

A good quick read, makes you realise how useful a rough and ready estimate can be, and how to get there. Definitely learned something reading this.

[David · Rating 3 out of 5]

Well presented, entertaining and "nifty". Nice to learn a few new things such as miles to km based on the Fibonacci rule and rough square root calculations

[Trish · Rating 5 out of 5]

A fun, approchable maths book! And believe me, that's not a sentence I ever expected to write.

[Mariela Stancheva · Rating 3 out of 5]

“Математика на гърба на плика: Умни начини да изчислите (приблизително) каквото и да било” е книга с много симпатични идеи защо математиката ни е необходима ежедневно. Авторът Rob Eastaway е избрал редица типични ситуации в които познанието на основни математически познания са важни за оценка на тези ситуации. За този пост съм избрала една задача, която трябва да присъства във всеки български учебник по математика. Защото от нейното разбиране зависи правилната реакция в ситуация, в която рано или късно изпада всеки човек. Ето и задачата:

“Дете тежащо 20 кг.има инфекция и се нуждае от лечение с антибиотик амоксицилин. Указанието е да се приема 25 милиграма амоксицилин за всеки килограм тегло на пациента на всеки 12 часа. Медикаментът пристига в опаковка с концентрация 250 милиграма на 5 милилитра. Колко (в милилитри) трябва да е препоръчаната доза за детето?”

Лекарите редовно решават тази задача наум. Това, което е важно е всеки човек също да може да се ориентира в такава з��дача. Това ще даде възможност на всеки да се докосне до истинската полза от математиката, но е и чудесно упражнение за използването на математика извън класната стая. Следващият път когато някой се р��зболее и отвори листовката на пациента, която задължително е част от всеки лекарствен продукт ще види там урока по математика, който спасява живот.

“Математика на гърба на плика: Умни начини да изчислите (приблизително) каквото и да било” съдържа и други интересни идеи. Научно-популярните книги по математика са такава рядкост, че всяка една е ценен принос към всяка библиотека.
Мисията на “Наръчник на бунтаря за проваляне на часа по математика” е да покаже практическите ползи за личен успех от изучаването на математика. Ясно е, че учебният материал е крайно недостатъчен и ние трябва да добавим още основни математически концепции към сега съществуващите учебни планове. Но и още нещо е важно – изучаването на сега включените в учебните планове математически концепции да бъде обвързано с тяхното използване в ежедневието. Това е много важно за мотивацията за изучаване на тези концепции. Сега в учебниците се появяват задачи, които са опростени ситуации без особена връзка с реалността. Вместо това ви предлагам да извадим подходящи задачи направо от действителността. При това нека те да са такива, че учениците да са се сблъсквали със ситуацията в която се появяват тези задачи.

[Philip · Rating 3 out of 5]

A good book on getting an intuitively good "feel" for numbers. This is an important and pretty fun topic and a good challenge to the "I just can't do maths" crowd.

The kind of thinking advocated in this book forms an integral part of my day job (I'm currently a physicist), so although there were a couple of fun tricks I hadn't come across, most of the topics in the book were already familiar to me. It perhaps didn't help that I read this shortly after reading David Spiegelhalter's The Art of Statistics: How to Learn from Data, which covered many of the same topics in a style that was more to my liking. I think this book was aimed at either adults who don't have much confidence with maths, or maybe a bright 10-year-old looking to do something fun with the arithmetic they mastered at primary school.

The layout of this book is perhaps a bit tedious - the final section was example after example of Fermi problems. And the writing felt a little dull: it was clear, but there was no "sparkle" or excitement to it. And the anecdotes and real-world examples in this book were all very UK-centric, which I found a little off-putting.

This was a pretty short book, so I'm surprised to find myself saying I'm glad it wasn't longer. But overall, a worthwhile, entertaining read.

[Joy · Rating 3 out of 5]

Disclaimer: I had to return this book to the library before finishing it, so this isn't a rounded review.

The first chapter or two were very good (maybe 4 stars?) with a good discussion of accuracy, precision and estimation and some nice tips for mental arithmetic.

Some of the examples later in the book weren't so good, and were based on false assumptions or bad sampling that could lead to very wrong estimations (even orders of magnitude out). I admit that I only read a couple of the examples and I read those quickly so they may not be representative of them all and I may have misunderstood some of their details. But estimating regular church attendance by the number of weddings and funerals seems nonsensical. And assuming that a sample of one YouTube video represents the typical American teenager also seems suspicious. As an academic, I know how conclusions drawn from bad samples can be very misleading.

So, overall, I think that the first few chapters are helpful but take the examples in later chapters with a big pinch of salt.

[Qothrunnada · Rating 4 out of 5]

4,5/5 ⭐️
This book is so good. Buku ini mostly isinya ngasih tips n trik ngitung matematika, in a fun way. Bukan ngajarin matematika, but it helps to see the bigger picture behind numbers. Tips perkalian cepat tanpa kalkulator yg bakalan berguna di kehidupan sehari2, mulai dari perkalian sederhana sampe ngitung diskon secara cepat. Trus entah knp ngerasa buku ini vibesnya mirip pengajaran Zenius lol 😂
Then, why not 5 stars? Cz ke bagian akhir kebanyakan ngasih pengaplikasian dan menurutku sedikit bikin bosan. But overall its good👍🏽

[Colin · Rating 5 out of 5]

This is very, very nicely done. I was expecting a bit of disappointment from this (I admire Rob Eastaway greatly, and I have Strong Opinions about mental arithmetic, so it's hard for a book to live up to my hopes) but on the contrary, it's delightful. The Fermi problems section is wonderfully done (good questions, interesting and challenging), and the tone is light and friendly throughout. A big recommend from me :-)

[IvanOpinion · Rating 4 out of 5]

This book should be compulsory reading for everyone. This is just common sense maths that everyone ought to understand, but probably less than 10% of us do.

I found that almost everything he suggests are things I already do, but then I trained in science and have worked in finance, so you pick these things up. But it was still interesting to have it all set out methodically and with excellent examples to show why his techniques matter to average people.

[Lily · Rating 4 out of 5]

I really enjoyed this! I did think the explanation (from the appendix) of what a significant figure is and how to find the first one should have been right at the start of the book, as I can imagine it might be a bit confusing otherwise.
It also helped me understand better the difference between arithmetic (which I had always previously seen as an antiquated term for mathematics) and mathematics. Turns out arithmetic is the bit I’m good at!

[Emil Petersen · Rating 3 out of 5]

The premise of the book is that the ability to estimate and make fast approximations is a great skill that should be cultivated. I agree. The first two or so chapters is about why this is good and what tools are readily available in most situations (besides a calculator). The majority of the book is actually examples and 'Fermi problems', i.e. silly interview-like questions such as "how many tennis balls are used at Wimbledon?'. It is a fast read with some good exercises in between chapters.

[Rob Sedgwick]

Entertaining

This is an interesting book. A lot of it I knew already, which quite pleased me. Although the idea of actually doing back of the envelope calculations when faced with a thought like "I wonder how many?" is definitely worth the encouragement this book gives you. I was lying in bed wondering how many dentists there were in Surrey the other day.

[Ulrich Schroeders · Rating 3 out of 5]

Hard to give a valid and fair review on this one. I'll do this back-on-the-envelope math quite often and think it's both fun and necessary in different real-life contexts. Did I learn something really helpful? Unfortunately, not. But the author is not to blame and did a good job on bringing a set of techniques to an interested readership.

[Vikrama Dhiman · Rating 4 out of 5]

Good Read

A good breezy read despite the mathematical problems in some cases to solve. I loved the chapter at the end with more amorphous problems like how much methane gas do cows emit daily. Recommend everyone who would love to do a refresher on market sizing quick math fundamentals.

[Daniel · Rating 4 out of 5]

There are many "Test Yourself" in this book with some explanations on how majority of ppl calculate the answers, and it's amazing to learn some "common" methods which are in fact not that quick. The square root shortcut is nicely presented and it really works! From one test, I have verified myself as arithmetician :)

[Devasya · Rating 3 out of 5]

This is a neat little book. It's interesting, and full of fun anecdotes, but I also find it slightly repetitive. I think anybody could pick this book up and enjoy it, even without a strong background in maths.

[Michal Kosinski · Rating 3 out of 5]

This book is solid 3.5. It is comprised of four parts: part 1: on importance of doing rough calculations; part 2: tips on how to calculate faster; part 3: simple everyday estimations; part 4: more advanced, Fermi questions.
As a maths MSc, I liked part 1, skimmed part 2, and enjoyed part 3 and 4.

[Harry Harman · Rating 5 out of 5]

Maths applied to real world examples is great and so refreshing.

Things like lottery numbers, estimating the height of a tree, the size of a crowd.

I wish this was the first book I’d pick up on maths.

[Ankit · Rating 3 out of 5]

This was a quick and easy read, with interesting anecdotes and some useful tricks. It is probably of more utility if you are used to calculators.
I'd recommend reading it for some fun exercises on estimation, but I felt not enough of the book was dedicated to that.