# My first mathematical discovery. What was yours?

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## Main Question or Discussion Point

This is occasioned by the thread
but being somewhat personal I thought the sometimes rather sobersided mods could find it out of place there.

I was first interested by the proposition that (n + m) = (m + n) (1) - I hope it doesn't sound like boasting - but I remember realising this theorem in a flash of inspiration, aged about 5! Believe me or not it was in an abandoned cowshed! I won't say I really lived up to this early promise. Actually I realised at first a special case, that 6 + 4 gave the same as 4 + 6 = 10 (2). No one had told me this! I had learned the parts of that as separate bits of information which the teacher taught with home-made bits of red paper cutouts that we fitted together with 1, 2,... squares ruled on each bit, so 10 was a 3X3 square with one extra little square hanging on the left bottom corner. It was only a couple of days before I realised the full generalisation of equation (2) above. (Other slight fascinations of this thing were the word 'plus' which was in those days not used for anything else, it was a somehow satisfactory word like 'plug' and goes well with the exceptional symbol which stood out from ordinary letters by its squat character in many fonts.)

Of course I was not the first person in the world to realise it, nor did I attempt a rigorous proof and fortunately I was never asked for one. On the other hand I was never given one either.

For that matter I wonder how many of our teachers realised it. Certainly for multiplication they seemed not to. In those days multiplication was taught be a recited drill called 'times tables' that certainly took up a lot of school timetable. We had to learn 2 times numbers up to 12X2 the 3X... up to 12 X 12.* We could have done adequately with learning just over half the amount. But if commutativity was mentioned it was at a later stag and then only as an incidental trick for slickly simplifying calculations and not by name, so I question whether my teachers at least at elementary school realised the importance of commutativity.

I don't know how they get the ideas across these days, but I know a lot of parents think kids should be taught as I and they were. Although they are pretty imprinted in me I have my doubts. In the last year of 'O' level English exams a set book was 'The History of Mr. Polly' and H.G.Wells mentions (the novel is amongst other things a bit of a critique of English education of his time) that Polly could never remember whether it was six sevens or seven eights that made fifty six - and had no way of finding out. That was an area of vague unreliability for me too though I am not as helpless as he. But if so I think it's my merit more than the school's!

*There was even a half hearted attempt to teach 11X and 12X, which was feasible because there are some simplifying features though I think these I vaguely intuited feather than was shown. I think we did get to 13X but at that time I detached myself. I think I had a feeling that up to 12X12 was some real but rounded knowledge that seemed to be demanded and that all numbers you could have been called upon to multiply ended there - if they did not then there was no other place to stop.

Doug Huffman
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I had a horizontal drum of heating oil for which I wanted a calibrated dip stick. While calculating the mapping of gallons to inches I noticed a recurrent term, the definition of radians.

PeroK
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I never really "learned" basic probability and combinatorics, as it all seemed to be in there to begin with! I remember finding a book of card games when I was about 13-14 and it gave the odds of being dealt each poker hand. So, I put the book aside and managed to work them all out for myself. I think I even got the "two pair" correct, which is the hardest, I think.

My first mathematical "discovery"--at age three--was that "three sevens is twenty-one". In other words, I was thinking about multiplication, but not explicitly--just in terms of adding a number to itself.
So, basically, in retrospect, I had found that a number multiplied by $n$ was like adding the number to itself $n-1$ times.

Fredrik
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...nor did I attempt a rigorous proof and fortunately I was never asked for one. On the other hand I was never given one either
It's an axiom, not a theorem. It's part of the definition of the real numbers. The commutative law is part of the definition because of what we want to be able to do with the real numbers.

There are lots of real-world physical things that we want to be able to calculate as 6+4. For example the number of kilometers you have run if you ran the 6 km track on Tuesday and the 4 km track on Thursday. Clearly this is the same number of kilometers that you have run if you ran the 4 km track on Tuesday and the 6 km track on Thursday. If the former is calculated as 6+4, then the latter should be calculated as 4+6. So if the definition of the real numbers doesn't make 6+4 equal to 4+6, then the method that we want to use to calculate distances wouldn't work.

Curious3141
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Can't pinpoint a *first* "discovery" with any degree of precision, but I can summarise some memorable insights.

When I was quite young (about 8 or so), the notion of trying to solve $x^3 - 5x = 5$ popped into my head. Mind you, I had just (autodidactically) learned basic algebra and I didn't know what a quadratic was, let alone a cubic. But I still persisted in banging my head against this problem until I met an adult who knew enough Math to tell me it was called a cubic equation. Then it was off to the encyclopedias where I learned about Cardano, Tartaglia and (Lodovico) Ferrari and their contributions to cubic and quartic solutions. I found the quartic solution very difficult, but the cubic was easy enough that I managed to solve this equation by hand, and I cannot tell you how much joy that gave me. :)

A bit later, I was playing around with multiplying rabbits when I game up with the notion of a geometric series (of course, I didn't know what it was called then). I tried summing it but was left with an unwieldy expression except for special cases like the powers of 2. A bit later, I had the rare good fortune to be visited by a distant relative and family friend called Gopal Prasad who showed me how to construct the closed form expression for the geometric sum, and also introduced me to Group Theory.

When I was young, I was also playing around with conjectures like "is p! + 1 prime"? (before I knew what a "primorial prime" was).

I self-taught myself calculus around the age of 11 or so.

• Z_Lea7
Mentallic
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Calculus at 11?? oo)

One of my earliest accomplishments that I can remember was being asked in Kindergarten (at a ripe young age of 4) how many days of the year there were. Answers were flying around from all across the room ranging from such extremes as 20 to the biggest number we knew - a million. I figured that since there were 30 days in a month, and 12 months in a year, I managed to multiply the two before I even knew what multiplication was. But of course, I was 5 short, so when the teacher said to add 5, the next kid that sprung his hand up got the answer and all of the praise to go with it.

So I learned from a young age that life isn't fair.

• Z_Lea7 and PeroK
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It's an axiom, not a theorem. It's part of the definition of the real numbers. The commutative law is part of the definition because of what we want to be able to do with the real numbers.

There are lots of real-world physical things that we want to be able to calculate as 6+4. For example the number of kilometers you have run if you ran the 6 km track on Tuesday and the 4 km track on Thursday. Clearly this is the same number of kilometers that you have run if you ran the 4 km track on Tuesday and the 6 km track on Thursday. If the former is calculated as 6+4, then the latter should be calculated as 4+6. So if the definition of the real numbers doesn't make 6+4 equal to 4+6, then the method that we want to use to calculate distances wouldn't work.
Perhaps detail of this needs to go back to the thread linked to in #1 - actually there are several themes arising - perhaps it can be an axiom, but briefly if it is an axiom why is someone in the link in #1 trying to prove it?

Excuse me if I don't try and settle that myself. I have dabbled in 'abstract algebra' which was some fun and enlightening, I seemed to get to roughly the same point as I did after 12X12! And let's say my time and even math learning priorities are such that I would mug it up only if and when I had to. :)

I certainly could have had no suspicion of a concept of real number. Fractions were in the unknown future. This was about natural numbers.
I think it was for me first an empirical finding about the tables we had learned, not only 6 + 4 = 4 + 6 = 10, but the same was true of 3 + 7 and 7 + 3 and so on. I realised, again I think in just a few days, that a similar rule held also when the total wasn't 10. I realised at least the special cases of 10 playing around either mentally or physically playing around with the cutouts, I could line up the 3 side of the 6 rectangle and the 3 + 1 cutout two ways, OK 4, and turn them around and they are exactly the same thing. Not only was but had to be. This physical manipulation must have the axioms built into it.

I case I have seemed to ungratefully disparage my teachers I should add that they had in those days neither lavish pre-supplied teaching aids, nor control freak educational ministers, and hierarchies of officials prescribing exactly what to do in their every lesson. The teacher, Miss Potts, had made, ruled and painted with watercolour all the cutouts for the whole class herself, I am sure of this as I can see them in my minds eye still now - a very belated tribute. :L

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I think the first one I can remember was maybe 6 years old or something.

I was going to grocery store for the first time alone. I had too little money for the things so it was annoying that's how it felt. I had to take less groceries.

I didnt think math rules made much sense then. I felt angry at the clerk because the rules didnt seem sensible to me. Clerk's rule sucked in my opinion.

Why would clerk's rule be better than my rule? I guess i didnt understand money too well back then...

Fredrik
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Perhaps detail of this needs to go back to the thread linked to in #1 - actually there are several themes arising - perhaps it can be an axiom, but briefly if it is an axiom why is someone in the link in #1 trying to prove it?
A number system can be defined either abstractly (like how we define terms like "vector space") or constructively (like how we define the specific vector space $\mathbb R^2$). The OP in that thread is working with a specific constructive definition of the natural numbers, which includes a definition of an addition operation. So his goal is to prove that the specific addition operation defined in his book is commutative.

Since you have dabbled in abstract algebra, you will probably understand the abstract definition of $\mathbb R$: First you define the term "ordered field" (this includes the commutative law). Then you define "Dedekind complete" in the following way: An ordered field is said to be Dedekind complete if every subset that's bounded from above has a least upper bound. Then you prove the theorem that says that all Dedekind complete ordered fields are isomorphic. Finally, you can define real numbers by saying that you're allowed to pick any Dedekind complete ordered field and denote it by $\mathbb R$ and call its elements "real numbers".

The only problem with this approach is that it's not obvious that a Dedekind complete ordered field exists (in the sense of ZFC set theory). If you want to prove this, you have to specify a set R, single out two of its elements that you will denote by 0 and 1, define two binary operations on R, define a unary operation on R, define a unary operation on R-{0}, define a strict total order, and then show that the set, the elements, the order and the operations together satisfy the definition of "Dedekind complete ordered field".

If you don't like the idea that there are many sets that can be thought of as "the real numbers", you can instead choose a specific existence proof and think of that as the definition of $\mathbb R$. This is the constructive approach. To someone who prefers the constructive approach, there's one preferred Dedekind complete ordered field that can be considered the field of real numbers. (One problem with this approach: When we define $\mathbb C$ as $\mathbb R^2$, with the obvious addition operation and a funny multiplication operation, we can't, strictly speaking, call a complex number with zero imaginary part a "real number").

The two approaches mentioned above are meant to define $\mathbb R$ in the framework of ZFC set theory. There's also the option of not defining them at all. This is a perfectly valid option. Then you don't say anything about what a real number is, what addition is, etc., and just write down a list of statements that will be considered true (e.g. "For all x,y, we have x+y=y+x."). This doesn't indicate a belief that real numbers "really" exist or that these statements are true in some absolute sense. We are simply choosing to study the branch of mathematics in which they are true. Those statements are of course called "axioms" in this approach too.

The reason why people mathematicians don't use that last approach is that in ZFC set theory, you only need to leave two things undefined: What a set is, and the meaning of the symbol $\in$ (i.e. what it means for a set to be an element of a set). If you can get by with only two undefined things, you don't want a bunch of other things left undefined.

HallsofIvy
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When I was 16 I took a summer course that involved "difference equations" (no one in the course had taken Calculus so would have no idea what a differential equation was) and some basic abstract algebra. I proved that the set of all solutions to a homogeneous linear difference equation formed a vector space.

OK, this is going to sound a bit sad, maybe even pathetic. When I first learned the multiplication table, I had no problem whatsoever, except when it came to learning the multiples of 9. No matter what I did, I honestly could never remember (and it didn't help that I hated math until the age of about 16). However, one day I kept thinking about it, and I realized something silly that many of you probably already know. I found that the multiples of 9 add up to 9. For instance, 9 × 2=18, 1+8=9; 9 × 3=27, 2+7=9; 9 × 4= 36, 3+6=9, etc.. When I got to 9 × 11, I almost thought that the same principle wouldn't apply, but it does :) (9 × 11 = 99, 9+9=18, 1+8=9). I know, its really simple, and I really should have figured it out long ago, but it has finally helped me remember the nines:p. However, I have yet to find another number for which the principle also applies.

PeroK
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OK, this is going to sound a bit sad, maybe even pathetic. When I first learned the multiplication table, I had no problem whatsoever, except when it came to learning the multiples of 9. No matter what I did, I honestly could never remember (and it didn't help that I hated math until the age of about 16). However, one day I kept thinking about it, and I realized something silly that many of you probably already know. I found that the multiples of 9 add up to 9. For instance, 9 × 2=18, 1+8=9; 9 × 3=27, 2+7=9; 9 × 4= 36, 3+6=9, etc.. When I got to 9 × 11, I almost thought that the same principle wouldn't apply, but it does :) (9 × 11 = 99, 9+9=18, 1+8=9). I know, its really simple, and I really should have figured it out long ago, but it has finally helped me remember the nines:p. However, I have yet to find another number for which the principle also applies.
It's also true for 3, in the sense that a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

And, a number is divisible by 11 is the alternating sum of its digits is a multiple of 11. E.g. 62546 is divisible by 11, because 6-2+5-4+6 = 11.

• Z_Lea7, TheDemx27 and DrClaude
.Scott
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The first one of any significance that I remember was when I was 8 sitting in class bored.
I had a piece of arithmetic paper and noticed that if I folded it in half, it was approximately the same proportions - but half the area.
Clearly, if I folded it again, it would be half the width and half the length of the original, and one quarter the area.
So I wanted to determine what the linear reduction factor would be for just one fold if the proportions stayed the same.
I realized that that factor multiplied by itself would have to be 1/2.

I didn't have any algebra or the procedure for finding a square root. But I had lots of time, so I figured it out pretty precisely.

• Z_Lea7
TeethWhitener
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It's also true for 3, in the sense that a number is divisible by 3 if and only if the sum of its digits is divisible by 3.
It's true in every base $n$ for the number $n-1$. So, for example, 483 is divisible by 7. Converting 483 to base 8 gives you 0o743 (or 743 base 8), and 7+4+3 is 0o16 (or 16 base 8 = 14 base 10), and 1+6=7 (........which is the same in base 8 and base 10).

As for my first mathematical discovery, the one that I can remember is when I finally figured out that 2 times 3 meant 2 repeated 3 times. That one blew my mind when I was on the playground one day in elementary school.

• Z_Lea7
Curious3141
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OK, this is going to sound a bit sad, maybe even pathetic. When I first learned the multiplication table, I had no problem whatsoever, except when it came to learning the multiples of 9. No matter what I did, I honestly could never remember (and it didn't help that I hated math until the age of about 16). However, one day I kept thinking about it, and I realized something silly that many of you probably already know. I found that the multiples of 9 add up to 9. For instance, 9 × 2=18, 1+8=9; 9 × 3=27, 2+7=9; 9 × 4= 36, 3+6=9, etc.. When I got to 9 × 11, I almost thought that the same principle wouldn't apply, but it does :) (9 × 11 = 99, 9+9=18, 1+8=9). I know, its really simple, and I really should have figured it out long ago, but it has finally helped me remember the nines:p. However, I have yet to find another number for which the principle also applies.
It gets even better!

For multiplication of 1 to 10 by 9, the tens digit is one less than the number (and the units place is easily computed 9 minus the recursive (repeated) sum of those digits, just as in your post), e.g. 6*9 = 54, 8*9 = 72, 10*9 = 90.

For multiplication of 11 to 20 by 9, the hundreds and tens digits, when taken together, form a number that is two less than that number, and the units place is easily computed as 9 minus the recursive (repeated) sum of those digits, e.g. 13*9 = 117, 20*9 = 180, 12*9 = 108.

For multiplication of 21 to 30 by 9, the hundreds and tens digits, when taken together, form a number that is three less than that number, and the units place is easily computed as 9 minus the recursive (repeated) sum of those digits, e.g. 23*9 = 207, 30*9 = 270.

...

and so forth. Can you see the pattern here? Can you generalise it? Now compute 12,345,678*9 without the aid of a calculator. You can use scratch paper, and it shouldn't take you more than half a minute. Last edited:
• Z_Lea7
TheDemx27
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I remember in elementary school we we learning probability, and we kept on getting problems where we had to list all possible sets of events, for instance in one bag you have x distinct marbles and in another bag you have y distinct marbles, and the only method we were given of solving the problem was to go through and list all of the possibilities. I thought about it a bit and I realized that I could just multiply x by y and get the total number of possibilities.

Also I arrived at a kind of "I think therefore I am" type of idea when lying in bed. I felt cheated when I heard it colloquially.

There were also a few more pertaining to physics but I can't remember them right now...

• Z_Lea7
As weird as this may sound but my first math discovery was literally when I finally figured out how to find the remainder in a division, at the same time, the decimal...at 18 years of age. I am 22 now.

The reason why is because I was one of those students who did nothing but complain about how hard math was until I got my act together and studied math more. It felt great to finally understand what I was incapable of when I was in the fifth grade. (Haha)

I came across harder equations but had more fun figuring them out, subsequently solving them.

I do not know if this properly answers your question, but I just had to let it out anyways.

• Z_Lea7