17

I remember being excited about the following at a young age. If you add consecutive numbers you get triangle numbers. Triangle numbers are fun. If you put two consecutive triangle numbers together you get a square number. You can also make a square number by adding the next odd number.


16

How about: Numbers go the other way, too (negative) You can cut numbers in half, forever What if you cut a number into three pieces? 1 million is a thousand thousands (100 is ten tens) If you don't know the number, call it a letter (or name it :) ) You can add letters together too What if you cut triangles in half? Rectangles? Can you make a box from ...


15

Recently, a student in my beginning algebra course offered the following to the class, regarding signed number multiplication: Assuming positivity is like love, and negativity is like hate, then... "If you love love, that's love." $\Rightarrow$ positive $\times$ positive = positive "If you love hate, that's hate." $\Rightarrow$ positive $\times$ negative = ...


11

Two good books that I liked when I read them years ago are Simon Singh: The Code Book. This is a great introductory book to cryptography. The book is not very mathematical heavy, but cryptography is very related to number theory, so I think the book works well and can function well as an inspirational book. Simon Singh: Fermat's Enigma. This is a book about ...


11

I tell students to visualize $<$ and $>$ as mouths. They always want to eat the bigger number.


10

I am a very big fan of The Number Devil written by a famous German author (normally known for his books about history or politics). The subtitle in the original German version is "Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben." (=A pillow book for all fearing mathematic.") It's a (nicele illustrated) story about a boy having fear of ...


10

William Dunham's Journey Through Genius is, ultimately, about a bunch of facts, but it's written very well and can be inspiring to a budding math student. How to Lie with Statistics is just a classic and deserves to be mentioned, even thought it's not really "math-heavy". The Kaplans' The Art of the Infinite is genuinely playful and seeks to present math ...


10

The minimax theorem states the following: Let $X\subset \mathbb{R}^{n}$ and $Y\subset \mathbb {R} ^{m}$ be compact convex sets. If $ f:X\times Y\rightarrow \mathbb {R} $ is a continuous function that is convex-concave, i.e. $$f(\cdot ,y):X\rightarrow \mathbb {R} \text{ is convex for fixed } y, \text{and}$$ $$ f(x,\cdot ):Y\rightarrow \...


10

Fatou's Lemma states: for nonnegative measurable functions $f_n$, $$ \int_E \liminf_{n\to\infty} f_n\;d\mu \le \liminf_{n \to \infty}\int_E f_n\;d\mu $$ The mnemonic is $$ \text{ILLLLLI}, $$ meaning "the Integral of the Lower Limit is Less than the Lower Limit of the Integral".


9

In terms of a book which inspires, I will offer Love and Math: The Heart of Hidden Reality by Edward Frenkel. It is an autobiography. It explains how a knowing mentor used physics to lure him into deeper math. I don't want to say too much and spoil it, but, he goes on to explain symmetries, groups, Lie groups, Loop groups, braid groups, Lie algebras, Kac-...


9

For a recent suggestion, check How Not to Be Wrong by Jordan Ellenberg. Lying in the "simple and profound" quadrant, the book also gives deserved attention to Condorcet, in addition to providing a very readable book for a wide audience. Rather than my saying more, let me direct you to some recent reviews/responses: LA Times Salon Scientific American WSJ ...


9

I highly recommend Paul Lockhart's Measurement


9

I strongly reccomend The Cauchy-Schwarz Master Class by J. Michael Steele. It could be read by advanced high school students who did well in calculus and have a strong interest in mathematics although it is probably better suited for first year undergraduate math majors. It reads like a novel that contains plenty of challenging exercises. Another book that ...


9

I think the short answer to your question is that you are (like many people) confusing the arithmetic that you called 'mathematics' at school with the mathematics that mathematicians do. I very rarely add up numbers above 10, or write down ones with several digits. I think I possibly do own a calculator... somewhere. I would suggest starting with some ...


8

The books by Willian Dunham ("Journey through genius" (Penguin, 1991), "Euler, master of us all" (MAA, 1999), "The calculus gallery" (Princeton University Press, 2008) are the ones I've read) are outstanding. They show mathemathics in terms of the original work (more or less, using modern notation), and motivate the subject matter well. They do require some ...


8

I greatly enjoyed Another Fine Math You've Got Me Into by Ian Stewart. Entertaining and at a pace that any level of mathematician or non-mathematician would be comfortable with, but nevertheless discusses some very interesting and beautiful topics.


8

I have about twenty kids' books listed at the books page on my blog, Math Mama Writes. If I had to narrow it down, my absolute favorite is probably The Cat in Numberland, by Ivar Ekeland, a five-chapter picture book dealing with the story of the Hotel Infinity (with Mr. and Mrs. Hilbert running the hotel, of course). The confused cat is charming, Ekeland's ...


8

Martin Gardner's The Colossal Book of Mathematics. It contains many of his best columns from Scientific American on recreational mathematics. My favorite chapter is the April Fool's day chapter, which includes a 'counter-example' to the Four-Coloring Theorem.


8

Surreal Numbers by Donald Knuth is a story about two people discovering the surreal numbers and proving theorems about them.


7

I'm not exactly sure what's being asked here, since you say you want to to learn how "to do advanced math," but also that you "want to try algebra and trigonometry, to play with equations." I can't tell exactly what you mean by do advanced math, but you say: I'm pretty good at understanding concepts like Cantor Sets, mathematical logic, etc. So: It seems ...


7

There is a great old Disney short called Donald Duck in Mathmagic Land. As well as being delightfully drawn in the traditional Disney style, it contains lots of useful and occasionally surprising information about where math can be found in everyday life. Personally I found it informative and inspiring, I imagine there would be plenty of conversations to ...


6

At the books page on my blog, Math Mama Writes, you can find a list that includes about a dozen great books at adult level. (Scroll down past the kids' books to see them.) If I had to pick my favorite, I think it would be Math Girls, by Hiroshi Yuki (along with Math Girls 2). The unnamed protagonist is a boy in high school who loves math. He helps Tetra ...


6

I look forward to the responses to this interesting question. Instead of answering directly, permit me to make an analogy to art. Frank Stella is a renowned artist whose influence and compositional talent cannot be questioned. Nevertheless, I happen to know from reliable second-hand reports, that he cannot "draw" in the sense that, say, Leonardo, or even a ...


6

The earlier question, "Teaching a very enthusiastic and bright 5 year old" could help. Building and manipulating shapes enhances geometric imagination. Consider polydrons, or snap-cubes:             Learning Resources Snap Cubes


6

Below are some great and inspiring books by an excellent mathematician. (In the Really Big Numbers book, on the page where counting by tens is discussed there is an inspiring error (?)…Big Bird is right, everyone makes mistakes!) Really Big Numbers and You Can Count on Monsters by Richard Evan Schwartz. Let me also add this wonderful activity JDH did with ...


5

You Can Count on Monsters, by Richard Evan Schwartz, creatively illustrates prime factorizations for the first 100 numbers. Some of my godchildren are very into it. Though I think most of the math content is going over their head at this point, they are into counting the dots, and they're into the idea that numbers have personalities.


5

There is a awesome list on similar topics on Maths Stack Exchange, even more awesome list on Math Overflow. I heard a lot of good things about Smullyan's bibliography, and a lot of bad things about Gödel, Escher, Bach.


5

Another book, to add to @Jon Bannon's list, that my 4 year old daughter and I can recommend is Introductory Calculus For Infants. We also seem to discuss Graham's number a bit after watching the Numberphile videos


5

Arranging counters into groups (multiplication and division), so arranging 12 counters into 6x2, 3x4 etc, and realising that there are some numbers that cant' be arranged, no matter how hard you try (prime numbers). Then, how many prime numbers are there, can you work out which number is going to be prime?


4

an old one, but still a good one Courant & Robbins, What is Mathematics? It explains what mathematics is all about, for someone who may not know, but is willing to spend the time to learn. In getting the link, I was delighted to discover that the latest edition is "Revised by Ian Stewart"


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