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Beyond Infinity: An expedition to the outer limits of the mathematical universe
The brilliant and charismatic Eugenia Cheng takes us on a dizzying journey through what's bigger than infinity and smaller than its opposite
Every child had a schoolyard fight that ended with this classic exchange: “Nuh-uh, times infinity!” “Yeah-huh, times infinity plus one!” The argument goes to the heart of a question everyone has wondered about: What is infinity?
Musician, chef, and mathematician Eugenia Cheng has some answers. Whether pondering why some numbers are uncountable, or why infinity plus one is not the same as one plus infinity, Cheng takes readers on a staggering journey from math at its most elemental to its loftiest abstractions. Along the way she considers how you could use a chessboard to help plan a dinner party for the world, what it would mean to make a chicken-sandwich sandwich, and if you could create infinite cookies from a finite ball of dough (the math says yes!).
For readers of Here's Looking at Euclid and The Joy of X, Beyond Infinity shows how this little symbol—∞—can make us kings and queens of infinite space.
Every child had a schoolyard fight that ended with this classic exchange: “Nuh-uh, times infinity!” “Yeah-huh, times infinity plus one!” The argument goes to the heart of a question everyone has wondered about: What is infinity?
Musician, chef, and mathematician Eugenia Cheng has some answers. Whether pondering why some numbers are uncountable, or why infinity plus one is not the same as one plus infinity, Cheng takes readers on a staggering journey from math at its most elemental to its loftiest abstractions. Along the way she considers how you could use a chessboard to help plan a dinner party for the world, what it would mean to make a chicken-sandwich sandwich, and if you could create infinite cookies from a finite ball of dough (the math says yes!).
For readers of Here's Looking at Euclid and The Joy of X, Beyond Infinity shows how this little symbol—∞—can make us kings and queens of infinite space.
304 pages, Paperback
First published March 9, 2017
231 people are currently reading
2709 people want to read
About the author
Eugenia Cheng
17 books338 followersEugenia 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.
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Displaying 1 - 30 of 157 reviews
January 10, 2020
Maths As Art Form
If you find it hard to tell your reals from your naturals or can’t remember why infinitely repeating decimals aren’t irrational, then this may be the book for you. A charming sortie into the poetry of mathematics, a guided tour of what they didn’t teach you in school: how numbers work and what it means to say that there are an infinite number of them. Cheng knows how to make a narrative with a beginning, middle and end, and with just the right touch of humour. Beyond Infinity is an enlightening, understandable, readable introduction to the abstract art of mathematics. You may or may not want to practice that art after reading it, but you will certainly have an appreciation for the excitement and beauty of mathematics that you didn’t have before.
If you find it hard to tell your reals from your naturals or can’t remember why infinitely repeating decimals aren’t irrational, then this may be the book for you. A charming sortie into the poetry of mathematics, a guided tour of what they didn’t teach you in school: how numbers work and what it means to say that there are an infinite number of them. Cheng knows how to make a narrative with a beginning, middle and end, and with just the right touch of humour. Beyond Infinity is an enlightening, understandable, readable introduction to the abstract art of mathematics. You may or may not want to practice that art after reading it, but you will certainly have an appreciation for the excitement and beauty of mathematics that you didn’t have before.
March 5, 2017
Popular maths writers have it much harder than authors of popular science books. Pretty well everyone loves science at junior school, even if they're put off it in their teens, so for science writers, it's just a matter of recapturing that childhood delight in exploring how the world works. But, to be honest, maths is a relatively rare enthusiasm at any age, so the author of a popular maths book has to really work at his or her task - and this is something Eugenia Cheng certainly does, bubbling with enthusiasm and trying hard not to put us off as readers as she explores infinity.
In Cheng's earlier book, Cakes, Custard and Category Theory, food played too heavy a role for me - here that tendency reigned in, though it still rears its head occasionally. We get a quite detailed exploration of infinity, infinitesimals and some additional material such as infinite dimensions and infinite-dimensional categories (we had to get some category theory), plus the usual enjoyment of quantum weirdness. I felt sometimes, because the book doesn't build on a historical basis, we were thrown in at the deep end a little too early with assertions like infinity+1=infinity and infinity x infinity = infinity. The process where Cheng shows us how infinity can't be a real number, or an integer, or a rational fraction etc. also felt a little repetitive and drawn out. There's always a difficulty in letting go a little when you work in a field that requires total precision. But we never lose Cheng's enthusiasm and light touch.
I think this book will particularly appeal to a reader who already has an interest in maths, but not much training, because it is purely focussed on the mathematics itself. For the more general reader, I suspect a book like A Brief History of Infinity, which gives historical context, brings and people and social implications to frame the maths, would work better as an introduction. With the appetite whetted, though, they should be encouraged to go onto Beyond Infinity, which as a wider mathematical context.
I really enjoyed this book, and though the author's desire to include food did still slightly intrude - and I felt it was just a bit too much about her, rather than the maths - it's a great addition to the relatively sparse popular maths shelf.
In Cheng's earlier book, Cakes, Custard and Category Theory, food played too heavy a role for me - here that tendency reigned in, though it still rears its head occasionally. We get a quite detailed exploration of infinity, infinitesimals and some additional material such as infinite dimensions and infinite-dimensional categories (we had to get some category theory), plus the usual enjoyment of quantum weirdness. I felt sometimes, because the book doesn't build on a historical basis, we were thrown in at the deep end a little too early with assertions like infinity+1=infinity and infinity x infinity = infinity. The process where Cheng shows us how infinity can't be a real number, or an integer, or a rational fraction etc. also felt a little repetitive and drawn out. There's always a difficulty in letting go a little when you work in a field that requires total precision. But we never lose Cheng's enthusiasm and light touch.
I think this book will particularly appeal to a reader who already has an interest in maths, but not much training, because it is purely focussed on the mathematics itself. For the more general reader, I suspect a book like A Brief History of Infinity, which gives historical context, brings and people and social implications to frame the maths, would work better as an introduction. With the appetite whetted, though, they should be encouraged to go onto Beyond Infinity, which as a wider mathematical context.
I really enjoyed this book, and though the author's desire to include food did still slightly intrude - and I felt it was just a bit too much about her, rather than the maths - it's a great addition to the relatively sparse popular maths shelf.
May 29, 2017
I started reading this book expecting to learn more about transcendental numbers. The first couple of chapters introduced the concepts, but after that things went downhill. The author then began offering variations on the same theme - example after example of limits, series and other mathematical functions which utilize infinity, but no further explanations of transcendental numbers. Also, some of her examples are attempts at putting real world faces on concepts and miss the mark by a wide margin - using a robot's multi-jointed arm's degrees of freedom as dimensions.
This entire review has been hidden because of spoilers.
July 25, 2025
Infinity for lovers of shoes and baked goods
I started this book by mistake. I heard a presentation by Eugenia Cheng at The Perimeter Institute for Theoretical Physics, in which she told us a little about Category Theory. Category Theory is a field of mathematics I have never studied, and her talk got me interested. So I thought I would read her book about Category Theory. That is something I may someday do, but today is not that day. Cheng's Category Theory book is The Joy of Abstraction: An Exploration of Math, Category Theory, and Life. If I had been paying attention, I would have realized that Beyond Infinity: An Expedition to the Outer Limits of Mathematics is not it.
But it's a short book, so upon realizing my mistake I thought, "Well, why not? It'll be a quick dip into Cheng's writing." So, I read it. I confess myself disappointed. Cheng is a very chatty writer -- she talks a lot about herself personally, and I just couldn't relate. For instance, this
I was honestly not terribly impressed by her explanations of difficult mathematical concepts. She didn't make them easier to understand than the usual more formal explanations. (In my opinion, Ben Orlin's books are a better starting point for a nonmathematical reader seeking to understand difficult mathematical ideas.)
There are two mathematical concepts discussed in Beyond Infinity that I was not previously comfortable with: ordinal infinities and Category Theory. Having read it, I still don't understand them. Now, Category Theory only comes up as an example of multiple dimensions -- Beyond Infinity is not intended to explain Category Theory in any depth, so it can't be fairly criticized for failing to do so. Ordinal infinities, however, are very much a part of what Beyond Infinity is about.
In summary, if you prefer to take your math with baked goods and plenty of shoes, this might be the book for you. For me, except for the math, it was like a book written by an alien. I understand that The Joy of Abstraction is more mathematician-oriented, so I haven't given up on that yet.
Blog review.
I started this book by mistake. I heard a presentation by Eugenia Cheng at The Perimeter Institute for Theoretical Physics, in which she told us a little about Category Theory. Category Theory is a field of mathematics I have never studied, and her talk got me interested. So I thought I would read her book about Category Theory. That is something I may someday do, but today is not that day. Cheng's Category Theory book is The Joy of Abstraction: An Exploration of Math, Category Theory, and Life. If I had been paying attention, I would have realized that Beyond Infinity: An Expedition to the Outer Limits of Mathematics is not it.
But it's a short book, so upon realizing my mistake I thought, "Well, why not? It'll be a quick dip into Cheng's writing." So, I read it. I confess myself disappointed. Cheng is a very chatty writer -- she talks a lot about herself personally, and I just couldn't relate. For instance, this
Let’s assume the cookies are circular and perfectly even. I was once criticized in public for assuming this about scones, and accused of using factory-made scones. For the record, I do make my own scones, and I’m perfectly aware that nothing on earth is perfectly round and even, but it’s a good enough approximation for a math discussionI do not make my own scones. I cannot imagine feeling compelled to defend myself against the accusation of using factory-made scones, or even using the verb "accuse" in this context. Elsewhere she speaks of counting her shoes as an approach to visualizing infinity. She mentions with incredulity acquaintances who have "only" four pairs of shoes. I imagine that Cheng includes these personal asides in an attempt to make herself more relatable. The effect on me, however, is the opposite of that -- she feels like an incomprehensible alien, except when she's talking about mathematics, when she becomes human.
I was honestly not terribly impressed by her explanations of difficult mathematical concepts. She didn't make them easier to understand than the usual more formal explanations. (In my opinion, Ben Orlin's books are a better starting point for a nonmathematical reader seeking to understand difficult mathematical ideas.)
There are two mathematical concepts discussed in Beyond Infinity that I was not previously comfortable with: ordinal infinities and Category Theory. Having read it, I still don't understand them. Now, Category Theory only comes up as an example of multiple dimensions -- Beyond Infinity is not intended to explain Category Theory in any depth, so it can't be fairly criticized for failing to do so. Ordinal infinities, however, are very much a part of what Beyond Infinity is about.
In summary, if you prefer to take your math with baked goods and plenty of shoes, this might be the book for you. For me, except for the math, it was like a book written by an alien. I understand that The Joy of Abstraction is more mathematician-oriented, so I haven't given up on that yet.
Blog review.
November 23, 2017
There are to day a lot of good popular books about mathematics that those of some age and fond of maths would like have read when young,this one is one of theese books.
Here Eugenia Cheng explains in a masterful way for the layman concepts of advanced mathematics,as the infinite series of infinite cardinals and ordinals,the concept of infinite countable sets is to say those that one can put in a one to one correspondence with the natural numbers and the uncountable sets where one cant ,as for example the real numbers ; in this theory developed by Cantor the first infinite cardinal are the countable sets or aleph-zero,the scond infinite cardinal is aleph-one the cardinality of real numbers.
Cheng also explains the paradoxes of the infinite sets as for example that there are so many even numbers as natural numbers,or that there are so many real numbers in the interval 0-1 as in the whole real line,cheng also demostrates the uncountability of irrationals numbers following the diagonal trick devised by Cantor and that betwwen to rational numbers ever there is a irrational number.
In the second part she makes clear the very abstract concept of categorie,a thing formed by objects and relations (morphisms),for example we have the category of vectorial espaces where the objects are the vectorial espaces and the morphims the linear funtions between spaces.The theory of categories was first developed by Eilemberg And Mc Lane in the 40s of XX century,its importance at first was not well understood an was dismised as "abstract nonsense".
In the third part Cheng touchs the subject of infinitely small quantities and numbers,the related Zeno paradox and this notion as the foundament of calculus, explaining the formal building of real numbers using the concept of limit grafically developed trought a series of smaller neighborhoods in the line.
The book ends with the subject of the relation between the armonic series,the sum of inverses of natural numbers ,and the natural logaritm ,giving way to a solid of revolution of finite volumen and infinite Surface.
A very good book about the Cantor theory of infinite sets ,the concept of categorie (the field research of Cheng ) and the structure of the real numbers.
Here Eugenia Cheng explains in a masterful way for the layman concepts of advanced mathematics,as the infinite series of infinite cardinals and ordinals,the concept of infinite countable sets is to say those that one can put in a one to one correspondence with the natural numbers and the uncountable sets where one cant ,as for example the real numbers ; in this theory developed by Cantor the first infinite cardinal are the countable sets or aleph-zero,the scond infinite cardinal is aleph-one the cardinality of real numbers.
Cheng also explains the paradoxes of the infinite sets as for example that there are so many even numbers as natural numbers,or that there are so many real numbers in the interval 0-1 as in the whole real line,cheng also demostrates the uncountability of irrationals numbers following the diagonal trick devised by Cantor and that betwwen to rational numbers ever there is a irrational number.
In the second part she makes clear the very abstract concept of categorie,a thing formed by objects and relations (morphisms),for example we have the category of vectorial espaces where the objects are the vectorial espaces and the morphims the linear funtions between spaces.The theory of categories was first developed by Eilemberg And Mc Lane in the 40s of XX century,its importance at first was not well understood an was dismised as "abstract nonsense".
In the third part Cheng touchs the subject of infinitely small quantities and numbers,the related Zeno paradox and this notion as the foundament of calculus, explaining the formal building of real numbers using the concept of limit grafically developed trought a series of smaller neighborhoods in the line.
The book ends with the subject of the relation between the armonic series,the sum of inverses of natural numbers ,and the natural logaritm ,giving way to a solid of revolution of finite volumen and infinite Surface.
A very good book about the Cantor theory of infinite sets ,the concept of categorie (the field research of Cheng ) and the structure of the real numbers.
April 4, 2019
I had nothing but the best intentions.
This book is the mathematical equivalent of asking a meteorologist why the sky is blue. When answered comprehensibly, the answer is much more than any simple patron would necessitate. Just say "because of tiny molecules in Earth's atmosphere," or something to that effect. Anything more would make me feel like a dummy.
So in my good intentions, I, a writer and wordsmith, wanted to broaden my mind by accessing the mathematical lobe left dormant after my freshman year of college. In doing so, I awakened a glaze upon my eyeball and a hopelessness of passing Calculus tests I no longer need to take.
As such, there are times when a book review is less about the successes or failings of the book, but more about the successes or failings of the reader. I will demonstrate the latter.
- Infinity is much more than a concept. The thought of limitlessness is incomprehensible. Just saying that space is infinite doesn't give us the gravitas of what that means. As it turns out, you can put that sideways 8 into a math equation and get actual results. I did not know this.
Furthermore, the rules expressed (1+Infinity vs Infinity+1, for example) garnered much more creativity than I would have ever thought. Hilbert's Hotel blew my mind, and it was probably the simplest concept here. These mathematicians were incredible thinkers, and I can't regurgitate a lick of their findings as it flies above my head far out of reach.
The one argument I do have is with the cookie logic: That if you make each cookie by using half the dough from the previous cookie, you'll never run out. At some point, you're getting into the Quantum Realm and there are some places we shouldn't tread. Same goes for the "to get anywhere, you need to cover half the distance, so can you ever get there?" logic. This is fun to discuss for smarty-pants', but at some point, you're figuratively splicing film with a NASA-grade microscope. It's absolutely unnecessary.
- You should know intermediate math upon reading this book. As previously alluded to, my last math class was 10 years ago. It was statistics, one of the easiest classes I could have taken. I never knew what a factorial was. The educational system failed me. So it's quite difficult to wrap your head around infinity's applications when you are learning new forms of math all the while.
- Get a physical copy of this book. I've read a book or two in my day and I'm sure a book like this had charts. And I'm sure seeing numbers and equations as opposed to having them read aloud to me would have been a much more effective way to consume the hard-hitting number puzzles presented to me. Instead, I opted for the audiobook and almost had multiple brain malfunctions on the highway.
If you are a number junkie and want a comprehensive look on the concept and applications of infinity, this is a great book. If you are merely trying to expand your brain by stepping out of your comfort zone, better start with a bicycle before you hop on a unicycle. Which, by the way, have circle wheels. Which, by the way, are infinite if you trace them.
This book is the mathematical equivalent of asking a meteorologist why the sky is blue. When answered comprehensibly, the answer is much more than any simple patron would necessitate. Just say "because of tiny molecules in Earth's atmosphere," or something to that effect. Anything more would make me feel like a dummy.
So in my good intentions, I, a writer and wordsmith, wanted to broaden my mind by accessing the mathematical lobe left dormant after my freshman year of college. In doing so, I awakened a glaze upon my eyeball and a hopelessness of passing Calculus tests I no longer need to take.
As such, there are times when a book review is less about the successes or failings of the book, but more about the successes or failings of the reader. I will demonstrate the latter.
- Infinity is much more than a concept. The thought of limitlessness is incomprehensible. Just saying that space is infinite doesn't give us the gravitas of what that means. As it turns out, you can put that sideways 8 into a math equation and get actual results. I did not know this.
Furthermore, the rules expressed (1+Infinity vs Infinity+1, for example) garnered much more creativity than I would have ever thought. Hilbert's Hotel blew my mind, and it was probably the simplest concept here. These mathematicians were incredible thinkers, and I can't regurgitate a lick of their findings as it flies above my head far out of reach.
The one argument I do have is with the cookie logic: That if you make each cookie by using half the dough from the previous cookie, you'll never run out. At some point, you're getting into the Quantum Realm and there are some places we shouldn't tread. Same goes for the "to get anywhere, you need to cover half the distance, so can you ever get there?" logic. This is fun to discuss for smarty-pants', but at some point, you're figuratively splicing film with a NASA-grade microscope. It's absolutely unnecessary.
- You should know intermediate math upon reading this book. As previously alluded to, my last math class was 10 years ago. It was statistics, one of the easiest classes I could have taken. I never knew what a factorial was. The educational system failed me. So it's quite difficult to wrap your head around infinity's applications when you are learning new forms of math all the while.
- Get a physical copy of this book. I've read a book or two in my day and I'm sure a book like this had charts. And I'm sure seeing numbers and equations as opposed to having them read aloud to me would have been a much more effective way to consume the hard-hitting number puzzles presented to me. Instead, I opted for the audiobook and almost had multiple brain malfunctions on the highway.
If you are a number junkie and want a comprehensive look on the concept and applications of infinity, this is a great book. If you are merely trying to expand your brain by stepping out of your comfort zone, better start with a bicycle before you hop on a unicycle. Which, by the way, have circle wheels. Which, by the way, are infinite if you trace them.
September 9, 2017
There are some big numbers out there, footballers earn a jaw dropping amount per year, for what I am not entirely sure… The global economy is around US$107.5 trillion, there are approximately seven quintillion, five hundred quadrillion grains of sand on the earth and it is thought that there are 10 times as many stars as that. All of these numbers are frankly huge, enormous, gargantuan even, but compared to ∞ they are a mere drop in the ocean. In this book, Eugenia Cheng takes us on a journey to the outer reaches of the mathematical universe to contemplate the slightly abstract concept that is infinity. In it she poses various questions about this number, asking if 1 + ∞ is larger than ∞ + 1, are some infinities larger than others, can you fit an infinite number of people in Hilbert's Hotel and when does a number start becoming irrational.
Thankfully this book has lots of diagrams as Cheng sets about explaining the concepts of infinity, from the very simplest right up to the most detailed. I found most of it straightforward, but occasionally it was fairly tough going. When trying to get your head around infinity has challenged mathematicians for ages so it is not going to be easy for us mere mortals. Cheng endeavours to keep the prose readable, however, someone who has not picked up a maths book since school might struggle with this, but most of the time she gets the concepts across clearly. Overall a good introduction to infinity.
Thankfully this book has lots of diagrams as Cheng sets about explaining the concepts of infinity, from the very simplest right up to the most detailed. I found most of it straightforward, but occasionally it was fairly tough going. When trying to get your head around infinity has challenged mathematicians for ages so it is not going to be easy for us mere mortals. Cheng endeavours to keep the prose readable, however, someone who has not picked up a maths book since school might struggle with this, but most of the time she gets the concepts across clearly. Overall a good introduction to infinity.
December 8, 2019
Generally speaking I totally liked the book, I don't have any specific things to complain about it. As the title suggests the book is about INFINITY, it starts from very basic concepts & aspects and gradually goes on to more complicated ones. I found it very interestingly split into a part 1 and a part 2.
Moreover, the author uses easy to understand vocabulary and a friendly tone. I felt like I was speaking with a very excited friend, who is sharing their passion with me. I think this is a very powerful aspect and I totally recommend you read this.
The author is well know for great comparisons between mathematical concepts and real life events. You get some lovely comparisons related to cooking (her favourite hobby) and travelling (which were some of my favourite ones... I am not a big fan of cooking). At this part, it might be that some comparisons are not really your type, but this is a very personal aspect. Anyway, there are some great explanations of complex mathematical concepts in here.
I have a more in depth review on YouTube, if you want to check it out: https://youtu.be/s7MYmCoyUv0
Moreover, the author uses easy to understand vocabulary and a friendly tone. I felt like I was speaking with a very excited friend, who is sharing their passion with me. I think this is a very powerful aspect and I totally recommend you read this.
The author is well know for great comparisons between mathematical concepts and real life events. You get some lovely comparisons related to cooking (her favourite hobby) and travelling (which were some of my favourite ones... I am not a big fan of cooking). At this part, it might be that some comparisons are not really your type, but this is a very personal aspect. Anyway, there are some great explanations of complex mathematical concepts in here.
I have a more in depth review on YouTube, if you want to check it out: https://youtu.be/s7MYmCoyUv0
April 12, 2017
This book was fascinating. I truly believe if I'd had a math teacher like Cheng I would have been fully engaged.
I loved algebra growing up, and I have fond memories of a friend of my mom who used to print off algebra equations for me to do while I was waiting for my mom to get off work. I'd anxiously look forward to them.
I look back at my time in school and wonder where the shift happened? It's frustrating for me as I am now so interested in physics, biology, and math, three things I avoided like the plague in school.
The idea of infinity would have never been something I'd have thought to read up on, but the way Cheng writes makes it all accessible while not feeling dumbed-down. I'm sure it gets more complex but I actually felt I was learning while also laughing along at all the anecdotes throughout the book.
Worth a read for sure whether you like math or not!
I loved algebra growing up, and I have fond memories of a friend of my mom who used to print off algebra equations for me to do while I was waiting for my mom to get off work. I'd anxiously look forward to them.
I look back at my time in school and wonder where the shift happened? It's frustrating for me as I am now so interested in physics, biology, and math, three things I avoided like the plague in school.
The idea of infinity would have never been something I'd have thought to read up on, but the way Cheng writes makes it all accessible while not feeling dumbed-down. I'm sure it gets more complex but I actually felt I was learning while also laughing along at all the anecdotes throughout the book.
Worth a read for sure whether you like math or not!
April 29, 2017
OK, I have understood I'm not interested in abstractions that give me nothing reusable in real world problems, even though I deal with some every day. Also, probably not for audioformat, I can't immediately get the English terms for math symbols, probably would have been better in text.
On the other hand, algebra is a must in the world today, and sooner children get the concepts, the better.
"I was helping two six year olds to understand symmetry." That killed me, haha, now I can understand why so much people or countries are backwards - most (too many?) of six year olds are parent conditioned into pray to imaginary gods, wait presents from Santa and still play with dolls/ trucks machines.
On the other hand, algebra is a must in the world today, and sooner children get the concepts, the better.
"I was helping two six year olds to understand symmetry." That killed me, haha, now I can understand why so much people or countries are backwards - most (too many?) of six year olds are parent conditioned into pray to imaginary gods, wait presents from Santa and still play with dolls/ trucks machines.
April 26, 2017
The premise of this book sounded interesting to me. Unfortunately the journey was not nearly as good as the promise. Eugenia Cheng obviously has a passion for mathematics but her enthusiasm cannot overcome the obstacles involved in making a complicated subject easy to follow. Ms Cheng seems to jump around quite a bit in trying to explain infinity and lost me several times. Your average layman would not find this an enjoyable read.
July 18, 2024
In Beyond Infinity, mathematician Eugenia Cheng guides readers to the puzzles, paradoxes, and implications of the concept of infinity. She uses category theory, functions, sets, number theory, and mathematical paradoxes like Hilbert's hotel and Zeno's paradoxes to illustrate the concept of infinity.
I loved how Cheng emphasized mathematics as a creative subject revolving around exploration and inquiry-we may not often associate math with creativity and inquiry. Her lighthearted analogies, conversational style, and inquiring voice made this an entertaining and captivating read. Sometimes her writing style went on tangents, discussing random personal details that some may find frivolous or irrelevant, but other than that, Beyond Infinity was overall a great introduction to the concept of infinity and the many implications it has. Readers may still want to conduct further research, especially if they want to learn the intricacies of higher-level concepts Cheng mentions, such as dimensions, category theory, and anything in the realm of calculus. Still, it was a pretty clear yet accessible introduction to those concepts. Some of my favorite ideas discussed in this book include Hilbert's hotels, the definition of a "dimension" and the possibility of infinite dimensions, and the uncountability of real numbers by using sets and Cantor's diagonal proof.
I loved how Cheng emphasized mathematics as a creative subject revolving around exploration and inquiry-we may not often associate math with creativity and inquiry. Her lighthearted analogies, conversational style, and inquiring voice made this an entertaining and captivating read. Sometimes her writing style went on tangents, discussing random personal details that some may find frivolous or irrelevant, but other than that, Beyond Infinity was overall a great introduction to the concept of infinity and the many implications it has. Readers may still want to conduct further research, especially if they want to learn the intricacies of higher-level concepts Cheng mentions, such as dimensions, category theory, and anything in the realm of calculus. Still, it was a pretty clear yet accessible introduction to those concepts. Some of my favorite ideas discussed in this book include Hilbert's hotels, the definition of a "dimension" and the possibility of infinite dimensions, and the uncountability of real numbers by using sets and Cantor's diagonal proof.
September 16, 2017
Eugenia Cheng likes to play with infinity. Among her favorite mind-boggling conundrums: If she were immortal, she could procrastinate forever.
But in a way, she writes, this book is not about infinity. It’s about a journey into the abstract unknown. This book tells how abstract thinking works and what it does for us. For Cheng, the usefulness of math is about mathematical thinking and how it sheds light on the thinking process.
“We use science to study the world abstractly,” writes Cheng. We use math to study science abstractly. Then, ultimately, we use category theory to study math abstractly. And by this point, Cheng wanders too far into the weeds for me. Three and a half stars.
Eugenia made an engaging and passionate appearance here in the spring. As it turned out, she appeared just as the bookshop celebrated its eighth anniversary. Daniel, the proprietor, printed out a full-size image of the figure eight, as a graphic symbol of the event. But then, always in the mood for some fun, he turned the 8 sideways, transforming it into ∞, the infinity sign, serving as the perfect segue to Eugenia’s talk plus q and a.
In my signed copy of her book, Eugenia dotted her “i” not with a dot but with ∞.
But in a way, she writes, this book is not about infinity. It’s about a journey into the abstract unknown. This book tells how abstract thinking works and what it does for us. For Cheng, the usefulness of math is about mathematical thinking and how it sheds light on the thinking process.
“We use science to study the world abstractly,” writes Cheng. We use math to study science abstractly. Then, ultimately, we use category theory to study math abstractly. And by this point, Cheng wanders too far into the weeds for me. Three and a half stars.
Eugenia made an engaging and passionate appearance here in the spring. As it turned out, she appeared just as the bookshop celebrated its eighth anniversary. Daniel, the proprietor, printed out a full-size image of the figure eight, as a graphic symbol of the event. But then, always in the mood for some fun, he turned the 8 sideways, transforming it into ∞, the infinity sign, serving as the perfect segue to Eugenia’s talk plus q and a.
In my signed copy of her book, Eugenia dotted her “i” not with a dot but with ∞.
July 1, 2021
The greatest challenge for any technical writer is to present a genuinely difficult topic to a general audience without resorting to hand waving, false analogy or outright nonsense. Cheng does exactly that. She writes clearly, intelligently and accurately about infinity, cardinal numbers, ordinal numbers and infinitisimals.
There was even an idea I hadn't met before. It is easy to count the shoes of a countable infinity of pairs, but to count the socks requires the Axiom of Choice.
There was even an idea I hadn't met before. It is easy to count the shoes of a countable infinity of pairs, but to count the socks requires the Axiom of Choice.
July 28, 2021
2.5 stars
October 9, 2022
After reading about infinity and the different types of infinities in 'Math without Numbers' by Milo Beckman, I decided to read a whole book about infinity. I did some search and discovered Eugenia Cheng's book.
The book is divided into two parts. The first part explores the concept of infinity in mathematics. We get to discover how we count, how we compare two collections of objects and find out which collection has more. Then we get to explore infinite sets and discover different types of infinity. On the way Eugenia Cheng shows us different sights like Hilbert's hotel with infinite rooms, and what happens when a chessboard is filled with rice grains, putting one grain on the first square, two grains on the second, and doubling the grains till the 64th square (the result is amazing and mind-blowing!) In the second part of the book called 'Sights', Cheng talks about the infinitely small and infinite dimensions and other places where infinity makes an appearance.
Eugenia Cheng's writing is charming and warm and filled with humour. It is also clear and accessible as Eugenia Cheng makes complex ideas easy to understand with real-world examples. She likes cakes and pies and they make frequent appearances in the story while discussing infinities.
I was surprised that I've read about most of the ideas covered in the book, like how we compare two collections, the different types of infinities, the Hilbert hotel with infinite rooms, and filling up a chessboard with rice grains – most of them in George Gamov's 'One, Two, Three... Infinity'. Just shows that I've been reading too many books on math. Probably need a change of scene.
I enjoyed reading Eugenia Cheng's 'Beyond Infinity'. Hoping to read more of her books. Have you read this book? What do you think about it?
The book is divided into two parts. The first part explores the concept of infinity in mathematics. We get to discover how we count, how we compare two collections of objects and find out which collection has more. Then we get to explore infinite sets and discover different types of infinity. On the way Eugenia Cheng shows us different sights like Hilbert's hotel with infinite rooms, and what happens when a chessboard is filled with rice grains, putting one grain on the first square, two grains on the second, and doubling the grains till the 64th square (the result is amazing and mind-blowing!) In the second part of the book called 'Sights', Cheng talks about the infinitely small and infinite dimensions and other places where infinity makes an appearance.
Eugenia Cheng's writing is charming and warm and filled with humour. It is also clear and accessible as Eugenia Cheng makes complex ideas easy to understand with real-world examples. She likes cakes and pies and they make frequent appearances in the story while discussing infinities.
I was surprised that I've read about most of the ideas covered in the book, like how we compare two collections, the different types of infinities, the Hilbert hotel with infinite rooms, and filling up a chessboard with rice grains – most of them in George Gamov's 'One, Two, Three... Infinity'. Just shows that I've been reading too many books on math. Probably need a change of scene.
I enjoyed reading Eugenia Cheng's 'Beyond Infinity'. Hoping to read more of her books. Have you read this book? What do you think about it?
July 28, 2019
Though its goal to be accessible for all readers of all types makes some passages slightly frustrating for first-year mathematics students (or people who know what subtracting is), Cheng succeeds at connecting many mathematical concepts through the mindblowing idea of infinity, while offering an intuitive explanation for many paradoxes and fields. For me, it was refreshing to read about many textbook subjects in such an open way, filled with examples. I hope this book takes away some of the fear many people hold for mathematics through Cheng's enthusiastic writing style.
December 16, 2019
For a while I felt like the book was going on too long, and my family made fun of me. What did you expect from a book about infinity, they teased. Well, in the end she does bring it around nicely -- the penultimate chapter is excellent -- but there are times when it drags a bit. Still I learned a lot from this and in Cheng's typical clear and fun style.
October 22, 2020
First, I think Eugenia Cheng would be a great person to meet in person, seems super nice and at times engaging in her writing-especially early in the book. However, after almost three hundred pages I do not feel like I have learned that much. Some interesting concepts were reviewed but in a meandering, flow of thoughts way which starts to have you asking, "are we there yet?"
I guess the question becomes is 1+infinite anecdotes = to infinite anecdotes +1 (this is a play on infinity plus 1 does not equal 1 + infinity page 127)? Hmmm? Here is a sample of one of the little yarns that litter the book like minefields and quickly morph from "cute" stuffed animal type yarns to tortuous, vicious Chucky doll ramblings as they start to kill your will to read on:
"I try not to count my shoes too often, as I might be shocked by how many pairs I have. My excuse is that my feet are really big and it's hard to find shoes that fit and that aren't ugly. When I was fat, my feet were even bigger and I got into the habit of buying every pair of shoes that fit, want's ugly, and wasn't too expensive. In those days not many shoes met those criteria, so buying them all was quite feasible. I basically had an algorithm for buying shoes, and it still meant that I only owned about for pairs of shoes when I graduated. (This is still more than some men I know.) When I lost weight, I discovered that my feet got smaller as well (I did not realize that feet could get fat). The difference was only about half a size, but this critical half size moved my feet from out of the standard range of women's shoes into the petite range. But I still had the same shoe-buying mentality, which meant I ended up buying quite a lot of shoes. I suppose as vices go, that's better than impulse-buying sports cars. Also maybe I'm protesting too hard. I now have to make decisions when I uy shoes, rather than just following an algorithm.
However, no matter how many shoes I buy…" page 89
Another:
"Let's assume the cookies are circular and perfectly even. I was once criticized in public for assuming this about scones, and accused of using factory-made scones. For the record, I do make my own scones, and I'm perfectly aware that nothing on earth is perfectly round and even, but it's good enough approximation for a math discussion! It's not exactly a life-and-death situation, after all." page 260
The self-deprecating and rambling anecdotes become tiresome. The book does present some cool ideas and as she says on page 12 "I'm going to start by playing around with the idea of infinity a bit, to free our brains…"
Cool idea, execution, in my reading of it, was not well done–the circuitous route she took could have been cut down by about a hundred pages and been tighter and better read.
I guess the question becomes is 1+infinite anecdotes = to infinite anecdotes +1 (this is a play on infinity plus 1 does not equal 1 + infinity page 127)? Hmmm? Here is a sample of one of the little yarns that litter the book like minefields and quickly morph from "cute" stuffed animal type yarns to tortuous, vicious Chucky doll ramblings as they start to kill your will to read on:
"I try not to count my shoes too often, as I might be shocked by how many pairs I have. My excuse is that my feet are really big and it's hard to find shoes that fit and that aren't ugly. When I was fat, my feet were even bigger and I got into the habit of buying every pair of shoes that fit, want's ugly, and wasn't too expensive. In those days not many shoes met those criteria, so buying them all was quite feasible. I basically had an algorithm for buying shoes, and it still meant that I only owned about for pairs of shoes when I graduated. (This is still more than some men I know.) When I lost weight, I discovered that my feet got smaller as well (I did not realize that feet could get fat). The difference was only about half a size, but this critical half size moved my feet from out of the standard range of women's shoes into the petite range. But I still had the same shoe-buying mentality, which meant I ended up buying quite a lot of shoes. I suppose as vices go, that's better than impulse-buying sports cars. Also maybe I'm protesting too hard. I now have to make decisions when I uy shoes, rather than just following an algorithm.
However, no matter how many shoes I buy…" page 89
Another:
"Let's assume the cookies are circular and perfectly even. I was once criticized in public for assuming this about scones, and accused of using factory-made scones. For the record, I do make my own scones, and I'm perfectly aware that nothing on earth is perfectly round and even, but it's good enough approximation for a math discussion! It's not exactly a life-and-death situation, after all." page 260
The self-deprecating and rambling anecdotes become tiresome. The book does present some cool ideas and as she says on page 12 "I'm going to start by playing around with the idea of infinity a bit, to free our brains…"
Cool idea, execution, in my reading of it, was not well done–the circuitous route she took could have been cut down by about a hundred pages and been tighter and better read.
October 24, 2017
In Beyond Infinity, Eugenia Cheng does a great job explaining different aspects of infinity. From the infinitely large to the infinitesimally small, Cheng helps the reader comprehend a variety of concepts that otherwise may have been challenging to understand. Cheng starts introducing infinity by defining it. She explains why it is not a number and goes through the steps that led to that conclusion. She concludes part one by showing that different infinities are larger than others and differentiating between ordinal and cardinal numbers.
In part two, she introduces different dimensions and describes how there can be infinite dimensions. She then proceeds to unravel the center of so many paradoxes, infinitesimally small things. After explaining the infinitesimally small, she concludes the book by discussing several paradoxes including Gabriel’s horn.
Beyond Infinity is a great read overall. The book, because of its recent publication date (2017), is very clear and easy to understand. To the interested reader, you should know that this book is not meant for everyone. If you have any interest in math though, I would definitely recommend it to you. I ended up loving it and so many mathematical concepts are much more clear because of it.
In part two, she introduces different dimensions and describes how there can be infinite dimensions. She then proceeds to unravel the center of so many paradoxes, infinitesimally small things. After explaining the infinitesimally small, she concludes the book by discussing several paradoxes including Gabriel’s horn.
Beyond Infinity is a great read overall. The book, because of its recent publication date (2017), is very clear and easy to understand. To the interested reader, you should know that this book is not meant for everyone. If you have any interest in math though, I would definitely recommend it to you. I ended up loving it and so many mathematical concepts are much more clear because of it.
June 30, 2017
Infinity is a confusing topic, and I think Ms. Cheng did a brilliant job of delivering it in an interesting, fun, engaging way. Yes, at times I didn't quite understand, but given the topic that's acceptable--I certainly understood more than I expected to, and that's all on her.
At the very least, I'm glad I don't have to move people from a one floor hotel with an infinite number of rooms, to a two floor hotel, each of which has an infinite number of rooms, and that's something prior to reading that I didn't realize I was grateful for.
(Note: 5 stars = rare and amazing, 4 = quite good book, 3 = a decent read, 2 = disappointing, 1 = awful, just awful. There are a lot of 4s and 3s in the world!)
At the very least, I'm glad I don't have to move people from a one floor hotel with an infinite number of rooms, to a two floor hotel, each of which has an infinite number of rooms, and that's something prior to reading that I didn't realize I was grateful for.
(Note: 5 stars = rare and amazing, 4 = quite good book, 3 = a decent read, 2 = disappointing, 1 = awful, just awful. There are a lot of 4s and 3s in the world!)
July 16, 2017
Two stars? This is a time when the rating tells you more about the reader than the book.
I thought I'd give a math book a try. I haven't done any math since high school. Let's see what happens if I go into it out of curiosity. The title sounds interesting.
Turns out I wasn't interested. I read really closely for the first several chapters, but had trouble giving a damn about any of it. There's a lot of talk about different infinities and trying to be precise about close differences between things and .... OK, it's all well-reasoned. OK, it's even well-written. OK, but .... I just didn't give a damn. I ended up just skimming the entire back half of the book.
I thought I'd give a math book a try. I haven't done any math since high school. Let's see what happens if I go into it out of curiosity. The title sounds interesting.
Turns out I wasn't interested. I read really closely for the first several chapters, but had trouble giving a damn about any of it. There's a lot of talk about different infinities and trying to be precise about close differences between things and .... OK, it's all well-reasoned. OK, it's even well-written. OK, but .... I just didn't give a damn. I ended up just skimming the entire back half of the book.
July 27, 2017
Starts well, then goes off the deep end.
February 6, 2023
It's just not written very well.
November 28, 2020
Lots and lots of personal anecdotes, lots and lots of repetition, not a lot of hard facts or mind expanding concepts.
March 12, 2019
I've always liked maths, and to be honest I really miss it in my current education. This book certainly doesn't reach the level that I have been used to think about concepts, but I loved it. Especially since infinity has always been a bit of an issue for me...
I loved Cheng's explanations and metaphors and talking about cake and cookies when actually conveying complex topics (even though sometimes I just wanted to tell her: "Stop calling yourself fat, your weight doesn't measure you"). I'm certainly no mathematician, but I dare say she managed to hit the spot between too complex and too simplified extremely well. I would have liked to hear more about all the things infinity affects, but I understand there's only so much space.
I'd love to read more books like this and read more about fascinating proofs and mathematical research altogether. I definitely recommend this book, especially for those not so friendly with maths. (But also those that are)
I loved Cheng's explanations and metaphors and talking about cake and cookies when actually conveying complex topics (even though sometimes I just wanted to tell her: "Stop calling yourself fat, your weight doesn't measure you"). I'm certainly no mathematician, but I dare say she managed to hit the spot between too complex and too simplified extremely well. I would have liked to hear more about all the things infinity affects, but I understand there's only so much space.
I'd love to read more books like this and read more about fascinating proofs and mathematical research altogether. I definitely recommend this book, especially for those not so friendly with maths. (But also those that are)
October 14, 2020
This book is easy to read, It felt like reading through a history narration rather than a book about some abstract math concept. The author uses short anecdotes here and there to explain the concepts to the novice reader. I liked this book except for the part it goes really slowly as if the reader is a small child or someone with no math background. It was also good to see how the author (a math professor) explains simple concepts that we skipped through when we were in elementary school like addition etc so thoroughly and how such basic building blocks build up to the big abstract concepts that intrigue even the professional mathematicians.
weird fact i learnt from the book, with a finite cookie dough on hand we can bake infinitely many cookies and line up these cookies to make a distance that is infinite.
weird fact i learnt from the book, with a finite cookie dough on hand we can bake infinitely many cookies and line up these cookies to make a distance that is infinite.
August 8, 2021
When I was told that I need to do a 50-minute session with 9th grade kids on the topic of "infinity", I thankfully remembered that Eugenia Cheng has written a book which I can pick ideas from.
Eugenia Cheng is such a great writer. She has a real talent for coming up with the most winsome examples and metaphors to communicate abstract ideas in maths. This book actually covers an impressive number of topics treated in undergraduate maths curricula (and even mentions some topics from graduate maths courses) but is pitched at the lay person. The chapters are very well-organized, and it's not the kind of book that people who actually know the rigorous maths would find annoying/misleading.
She also does an incredible job at capturing what pure mathematicians (and children) love about maths. And, as a bonus, the book is peppered with charming anecdotes from her life, and various cultural references (such as queuing up for a BBC Proms concert at the Royal Albert Hall) - but, incredibly, always in the service of elucidating a mathematical idea.
I want to be her friend and do all the cool things she does! :)
Eugenia Cheng is such a great writer. She has a real talent for coming up with the most winsome examples and metaphors to communicate abstract ideas in maths. This book actually covers an impressive number of topics treated in undergraduate maths curricula (and even mentions some topics from graduate maths courses) but is pitched at the lay person. The chapters are very well-organized, and it's not the kind of book that people who actually know the rigorous maths would find annoying/misleading.
She also does an incredible job at capturing what pure mathematicians (and children) love about maths. And, as a bonus, the book is peppered with charming anecdotes from her life, and various cultural references (such as queuing up for a BBC Proms concert at the Royal Albert Hall) - but, incredibly, always in the service of elucidating a mathematical idea.
I want to be her friend and do all the cool things she does! :)
January 11, 2025
I never thought much about infinity before. It was always just "forever" or "uncountable" or "bigger than the biggest thing I can think of" and that was all I needed it to be. Well, that is *still* all I need it to be, but this was still a really neat way to look at infinity.
Math has always been something I was decently good at, mostly enjoyed, and didn't plan to use much in my career. So a lot of the things that are discussed in this book are things I have briefly studied (calculus), heard about (set theory), or could at least wrap my head around (Hilbert's Hotel). I don't know that I'll ever use any of what I learned listening to this book (and I'm not sure yet how much I'll retain) but it was very cool when I understood how it was possible to prove that one infinity could be larger than another infinity. (No, I can't explain it to you. Sorry.)
While the narration was enjoyable and easy to listen to, there were many times (especially in the later parts of the book) where I could feel myself losing a grip on what was being discussed. I didn't have anything to hold on to, so to speak. I couldn't visualize what was being discussed (the way I had been able to earlier) and so I know there are parts that I didn't understand at all which I might have been able to grasp if I was reading the print version.
Math has always been something I was decently good at, mostly enjoyed, and didn't plan to use much in my career. So a lot of the things that are discussed in this book are things I have briefly studied (calculus), heard about (set theory), or could at least wrap my head around (Hilbert's Hotel). I don't know that I'll ever use any of what I learned listening to this book (and I'm not sure yet how much I'll retain) but it was very cool when I understood how it was possible to prove that one infinity could be larger than another infinity. (No, I can't explain it to you. Sorry.)
While the narration was enjoyable and easy to listen to, there were many times (especially in the later parts of the book) where I could feel myself losing a grip on what was being discussed. I didn't have anything to hold on to, so to speak. I couldn't visualize what was being discussed (the way I had been able to earlier) and so I know there are parts that I didn't understand at all which I might have been able to grasp if I was reading the print version.
April 11, 2022
Matematikte, binlerce yıl insanların aklını karıştırmış bir konu olan sonsuzlukla ilgili güzel ve basit bir kitap. Eugenia bu konuyu sanki Bilal'e anlatmış. Özellikle sonsuz büyükler konusunda hemen hemen hiçbir şeyi atlamamış ve olabildiğince açıklamış. Sonsuz küçükler kısmını biraz daha kısa tutmuş, daha doğrusu daha çok konuya daha kısa değinmiş. Bence iyi de yapmış. Sonsuz küçükler sonsuz büyüklere göre anlaması daha zor konular bence.
Bu kitabı matematikte sonsuzluk kavramı ile ilgili bir şeyler okumak isteyen herkese önerebilirim.
Bu kitabı matematikte sonsuzluk kavramı ile ilgili bir şeyler okumak isteyen herkese önerebilirim.
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