Profile Books Ltd Unequal The Maths of When Things Do and Dont Add Up.

by Eugenia Cheng

At first glance, the concept of equality in maths seems unambiguous. When we see the equality sign, we think of 'solving for x' or balancing two sides of an equation or maybe even the many famous equations that make use of this elegant, innocuous symbol.

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But between those parallel lines lies a mathematical playground of choice and abstraction, leading to far greater insight than you could have dreamed. As it turns out, sameness and difference, equality and inequality, are not nearly as straightforward as they seem.

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Unequal explores the rich and rewarding interplay between sameness and difference, from numbers to manifolds to category theory and beyond in a glorious celebration of mathematics that will change the way you look at maths - and the world around you - forever.

Genres: Nonfiction, Mathematics, Science

400 pages, Hardcover

Published June 5, 2025

About the author

Eugenia Cheng is a mathematician, pianist, and lecturer. She is passionate about ridding the world of math-phobia. Eugenia’s first book, How to Bake Pi, has been an international success. Molly’s Mathematical Adventure is her first children's book.

Reviews

[Stefano Lusoli · Rating 4 out of 5]

Thank you Jacqueline for recommending this one, and happy birthday to you!

"Mathematics isn't a series of rules, facts or answers. It's an invitation into a way of thinking. Welcome in! I'm glad you're here."

"This might seem obvious, or it might seem extremely obscure (which is the fate of much of mathematics - seeming obvious to some people and obscure to others). [...] I think it is actually simultaneously obvious and obscure, which is, to me, a wonderful thing about mathematics."

"I also wanted to include it even if you don't understand it because the act of doing maths involves a lot of not-understanding things. It might seem that people who are good at maths are those who understand it all easily, but in the end I think that to make progress in maths one needs to get used to not understanding things but sitting with them anyway. I think a better indicator of your future success at maths is not how quickly you can understand something, but how long you can spend with something you don't understand."

"More dimensions lead to more nuance, and more nuance leads to us being able to communicate things with more detail, and thereby understand each other more deeply."

"A great deal of mathematics turns out to be about bringing into consciousness things that we are doing intuitively. Perhaps it's all a form of intellectual self-awareness."

[Harry · Rating 3 out of 5]

For me there is a sweet spot for science and maths books that can best be summed up by Roger Penrose’s Road to Reality. This type of book lies somewhere between an introductory text for undergraduate students and a popular science book that treats readers as able to grasp the necessary maths and physics while still being accessible to a wider audience. I came across Unequal by Eugenia Cheng in my local bookshop a couple of weeks back, and after a short browse revealed that it falls roughly into this category, I decided to give it a go.

Unequal is a dive into the world of category theory in mathematics, which is the area of maths that examines mathematical structures themselves. The author has written a number of books that infuse everyday life with esoteric mathematical concepts – her book on infinity (Beyond Infinity) is one of the best popular maths books I’ve read. However, of late her books have tended to be peppered with illustrations of real-life applications of maths that at times veer into leftist dogma which I think has put off a lot of her readers – including me. While this tendency is to some extent true of Unequal, it tails off towards the end of the book and only rises to the level of irritation in a few places (be prepared for some sweeping statements about white men).

True to its title, Unequal presents its material in a somewhat unequal way – some earlier chapters labour simple points, while later chapters feel a bit rushed, perhaps because Cheng thinks the material is too advanced and anything more than an overview would lose readers. Add to this the tendency of earlier chapters to spend too much time trying to relate concepts from category theory to real life – often unsuccessfully in my view – and I think the book could have been better, hence three stars.

There is one howler in the book that I think should be pointed out. Cheng states (on page 10 of the hardback edition) that road signs painted on road surfaces are elongated as, when they are seen from the perspective of drivers driving at speed, “the surface shrinks in the direction of travel according to how fast they’re moving”. Not true unless Cheng drives near the speed of light!

[SY NA · Rating 3 out of 5]

This book falls into the vein of 25 Rocks which I read earlier this year. If i was being completely honest with myself, i'll put this in the "didn't finish" category because even after I finish the book, I cannot tell you the main thesis of the book other than what you can see on the front or back cover of the book.

The chapters in this book always begin with a math equation of some sort, and the grand thesis is really that not everything about math has to be about equality which we kind of implicitly assume when we see x + y = z. The book then takes the math equation or concept and then starts broaching real life topics with it, like equality doesn't have to be equal, such as when in a couple, one person prefers to cook and another prefers to clean, it doesn't make sense to force each person to equally clean and cook the same amounts. We deem it equal that they can share chores in a way that's beneficial to all and wave away the inequality that one thing or another takes up more effort.

This is fine for relatively simple concepts, but when she starts talking about harder concepts, it just loses me completely. this was not for a lack of concentration on my part, the authoress, who is also the narrator, acknowledges in parts of the audio book that listeners might just have to live with the fact that they cannot see equations and to check the enclosed pdf if they want to follow along.

I might end up checking out the dead tree version of this book at some point, but simply put, I think this is kind of the "math's version of philosophy", using math concepts to express the authoresses' philosophy. As these books goes, how well you react to it will depend on how your worldview lines up with the authoress. Mine lines up well so I think i put up till the end of the book with it because what she does say about equality in plain english is worth listening to. its maybe a stretch as to how you can use math equations or concepts to justify it, but hey, it takes all sorts.

Worth it if you're into math and want to see how stretchy it really is when it comes to asserting world views. =)

not recommended unless you are really into math.

[Austin · Rating 5 out of 5]

It's exactly what the title says: an exploration of the concept of equality across different mathematical domains. It's explained well and enthusiastically; those without an advanced math background will still learn a lot. However, the sections on category theory and homotopy theory are suddenly much more advanced than the surrounding material, and will probably confuse those not already familiar with the topics, or at least haven't read Cheng's other book, The Joy of Abstraction. It makes sense since this book also focuses on abstraction and developing mathematical maturity, something Cheng does really well. Overall it's an accessible book that's easily recommendable.

[Bob · Rating 3 out of 5]

Cheng writes engagingly about what makes the study of mathematics interesting, and important, to her, and how it can help us gain insights in our everyday social world. Much of the formalism raced over my head, and the index was not particularly helpful.

[S. · Rating 4 out of 5]

This book feels like the first couple weeks of lectures of an introduction to the philosophy of mathematics course. interesting and well written and reasoned and very logical, not groundbreaking.

[Richard Anderson · Rating 5 out of 5]

Thought journey through places I never thought I’d even think about it – namely category theory, and similar fields. Didn’t understand at all, but worth the journey.