New answers tagged proofs
3
This is a great question. Love it love it love it!!
The following is just what occurs to me off the top of my head.
Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as ...
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I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion.
I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly:
(a) Don't let on that you are going to prove the Pythagorean Theorem -- don't ...
0
But this leap (replacing $\sqrt{x}+2$ with $2$) seems so unnatural to students who are encountering this for the first time.
Students will encounter it eventually, for example in their next calculus course someone will ask them
Does this series converge?
$\sum_{n=1}^\infty \frac{1}{\sqrt{n} + 2}$
and a solution might start with the fact that
For $n \geq ...
1
I don't know whether this is a comment or an answer.
There is a bigger picture to keep in mind, which is that, beyond developing the students' understanding of the epsilon-delta definition and their ability to find delta, we also want to develop the students' cleverness. There's a certain art to finding the unnatural leaps that enable someone to make the ...
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You are answering your own question. But also keep in mind a bigger picture. The purpose of introducing the $\epsilon$-$\delta$ definition of continuity is not to make people proficient in finding a $\delta$ for a particular situation like $f(x)=\sqrt{x}$. Rather, the purpose is to develop appreciation of the need for a formal definition of continuity. The ...
3
Most of the answers assume, as does the OP, that the purpose of proof is verification. In fact "establishing the truth of a proposition" is only one of the reasons that mathematicians prove things.
The most-cited paper on this subject is probably Michael de Villiers' "The Role and Function of Proof in Mathematics" (Pythagoras, 1990, pp. ...
1
Now, he asked me why you'd need proofs for anything ever, since it's
enough to see that an equation works after testing it for a few values.
So let me give my take on a practical example which does not involve numbers or formulas, so it should be relative simple to understand.
Having a mathematical proof for some calculation is like having a map.
Let's say ...
1
There are really several distinct questions tangled up in this, so let me try to tease them apart.
The first question is, Are there people who are intrinsically incapable of creative work in mathematics?
The second question is, Are there people who eventually become incapable of creative work in mathematics?
The third question is, What evidence exists for ...
2
Let me be provocative.
Your friend is right. Nobody needs proof except mathematicians. The reason you want absolute proof is that you have a mathematical mindset. For all other people (farmers, carpenters, engineers, physicists) overwhelming evidence is sufficient.
It is telling that most non-trivial statements about actual nature cannot be proven to be ...
3
If you've got a sum you're keeping track of (e.g. you're adding up your purchases in a shop), you can quickly check whether your final answer is right by just repeatedly adding up the digits, ignoring any 9's, until you get a one-digit number, and then checking whether the sum of the digits of your sum is equal to the sum of the digits of the summands after ...
1
Keep this person as a friend and admit -- if asked -- that you have no good explanation for someone unfamiliar with mathematics. (In general, spouses of mathematicians never ask!) If your friend plays bridge or chess, discuss the strategies of the game. You could then say that proofs give mathematicians the guarantee that their strategies will work. ...
1
It is possible to tile (tessellate) a plane with any triangle.
It is possible to tile a plane with any quadrilateral.
It is possible to tile a plane with some pentagons (such as 'houses' where the sides all have equal length, the base is a square, and the top is an equilateral triangle).
It is possible to tile a plane with some hexagons (such as regular ...
6
Put 100 red marbles in a box and 1 white marble. When you blindly pick a few marbles and they are all red, it does not mean all marbles in the box are red.
0
Money, even imaginary money is a good motivator
Much mathematical thinking historically, came from gambling and card playing. If you want to avoid losing your shirt, you need to understand the mathematics. It is a very practical use of the subject.
A simple experiment is to throw two dice. Ask your friend to bet on which sum will come up most often (use a ...
2
Here's a surprising fact I've learned today on this very site:
Have your friend raise with a calculator
$1.5^{1.5^{...}} $.
At some point it goes to infinity.
Now let him do the same with
$1.4^{1.4^{...}} $
Funnily it stop growing. That raises some questions about the previous one : did it actually stop growing? Or was just the calculator screen to small and ...
1
since it's enough to see that an equation works after testing it for a few values.
Would they be willing to bet their life on that if that equation was being used for an autopilot in an airplane or some other piece of equipment? Or if the equation takes millions of possible combinations of values?
Suppose I make an engine. I test it and it works. Great. ...
0
Does your friend get bothered by multiplication? You say $+$ and $-$ are okay but division is problematic, so you skipped over multiplication.
If your friend is okay with multiplication, and knows about breaking up numbers into prime factors, here's something to claim: every number has at most three different prime factors ($45 = 3 \cdot 3 \cdot 5$ has just ...
2
There is quite an amazing collection in response to the
11 yr-old MathOverflow
question, Examples of eventual counterexamples.
(This a precursor to the already cited 9 yr-old MSE question,
Examples of patterns that eventually fail.)
Here's one (among many) I like (posted by Gerry Myerson)
The numbers $12$, $121$, $1211$, $12111$, $121111$, etc., are all ...
4
get him to draw a triangle (maybe even freehand), and measure the internal angles.
point out all the things in the world built with triangles - electricity pylons, eiffel tower, maps etc - triangles are important, and its important in order to build our world to understand triangles completely.
what was the sum of the internal angles? (in my experience it ...
1
To some extent, your friend is totally right. If you want to check if an equation always holds, then just checking several values is often a nice and practical way to do that.
But proofs can certainly come in handy sometimes. Here's a simplified version of a problem I encountered at work this week.
We had 8 cables of various lengths and 8 conduits of various ...
6
An example that directly contradicts the idea of "seeing that an equation works after testing it for a few values:"
Theorem (not): The polynomial $x^2-x+41$ is prime for every integer value of $x$.
This works after testing it for every integer $x = 0, 1, 2, \dots, 40$.
So "obviously" (according to your friend!) its value is a prime number ...
15
Most answers describe proof as an improved correctness checking tool.
But, I don't think this gets at the core of the issue. The goal of math isn't to check off theorems, but to understand them and consequently nature (broadest possible meaning of the word).
Proofs are a way to describe insights into a subject in a clear manner. Impossibility proofs tell ...
4
Try solving Pell's equation by trial and error. For example:
$x^2-61y^2=1$ has a whole-number solution $x=1$, $y=0$. Are there any others?
Try a few values for $x$ and $y$, and it will appear not. Is that a proof that there are no more solutions?
It turns out that the next smallest solution is $x=1766319049$, $y=226153980$, which would take quite a while ...
11
Maybe you can use some non-mathematical examples...
Black swans. Europeans assumed that all swans were white, because all swans in Europe are white, to the extent that a black swan was considered impossible--like a "flying pig". When they finally got to Australia and discovered that there are black swans, it was a big shock.
Suppose someone tells ...
9
Because mathematicians want to be really, really certain.
There are lots of ways you can be convinced that something is true, and be wrong. By requiring proofs before you are convinced, you can make that amazingly less likely.
A proof requires that someone lay bare all of the weaknesses in their claim, and show how those weaknesses are not a problem. ...
13
One thing that can't be shown without proof techniques is the impossibility of something. One reasonably concrete example of this is the impossibility of constructing a square and a circle with the same area using a straight edge and a compass. Individual cases can be shown to not work and to demonstrate that it's a hard problem, but there's a world of ...
5
From a practical point of view, your friend is right. One just has to be careful to test enough numbers. Engineers don't believe calculus because it has proofs based on analysis which is based on set theory; engineers believe calculus because lots of engineers before them used calculus to build bridges and very few of the bridges fell down. We have ...
30
More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. ...
13
Theorem: Suppose you start with the number 1, and you're allowed to multiply it by 2, 3, and/or 5, as many times as you like. In this way, it's possible to get any whole number.
Proof: Check this for 1, 2, 3, 4, 5, and 6. Also true for other random examples I think of, such as 10, 100, and 96. It's clearly true.
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I often teach a short unit on induction in elementary calculus classes—it comes up very naturally when trying to prove a version of the power rule:
Lemma 1: (Product Rule) Let $f$ and $g$ be real valued functions on $\mathbb{R}$. At any point $x$ where both $f$ and $g$ are differentiable,
$$ (fg)'(x) = f'(x) g(x) + f(x) g'(x). $$
As it is not really ...
0
It is obvious that the algorithm given in the accepted answer (in the Mathematics Stack Exchange Site) to the question “Divide a plane with 2n points into two equal halves” (at: https://math.stackexchange.com/questions/3083344/divide-a-plane-with-2n-points-into-two-equal-halves/3083372#comment8100957_3083372) can be slightly modified to fit the situation you ...
0
I usually explain this by simulating, and then defusing, the mis-understanding using a simple table. You can set up any of the three cases of the problem, as they are obviously all the same, I'll use [1:Goat] [2:Goat] [3:Car] here:
My choice Monty Shows Reward for Stick Reward for Switch
1 (goat) 2 (goat) 0 ...
2
How about encapsulating the entirety of the explanation in a tree diagram, which is visual, accessible, and relevant to any Intro Probability classroom?
We drew this today while going over the Monty Hall problem to cap off the topic (Probability) by considering interesting / counter-intuitive problems like the Birthday Problem, medical tests and the Sally ...
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