# You Don't Say?

Sometimes there are many ideas in math that make you go

Alt text

This note is going to be all about the most obvious facts about math. Post the ones that you think are extremely obvious and vote up your favorites.

I am adding a few to get things moving.

Have fun and be totally obvious!

For those of you who want a link to the image - https://i.imgur.com/Fvnb0Xl.png.

Note by Mursalin Habib
6 years, 2 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Jordan Curve Theorem

Every closed non-self intersecting continuous loop on the plane divides it into an interior and exterior region, i.e. inside and outside the loop. Proofs of it are some of the longest, most difficult ones in mathematics, usually involving algebraic topology.

Axiom of Choice

Given any number of non-empty sets, it's possible to pick exactly one member from each of those sets, which makes for a new non-empty set. That is, given a lot of boxes, all of them with things in it, I can get another box, and pick one item from all the other boxes and put them into this box. No proof exists for this one, so it has to be made into an axiom.

- 6 years, 2 months ago

I wish I could upvote you again and again! This is gold!

- 6 years, 2 months ago

$\displaystyle 0\times \pi\times \tau\times \phi\times e = 0$

$\displaystyle\text{MIND = BLOWN}$

- 6 years, 2 months ago

The Pigeonhole Principle

If you have things stuffed in containers and if there are more things than containers, then at least one container has more than one thing in it.

Imgur

- 6 years, 2 months ago

PHP wins in my opinion :D

- 6 years, 2 months ago

agreed

- 6 years, 2 months ago

agreed

- 6 years, 2 months ago

Reflexive property of equality

$a=a$

Imgur

- 6 years, 2 months ago

The obvious one is

$1 + 1 = 2.$

ON page 357 of Principia Mathematica, they finally conclude

From this proposition it will follow, when arithmetical addition has been defined, that $1+1=2$.

Staff - 6 years, 1 month ago

And then came along Kurt Godel.

- 6 years, 1 month ago

$\displaystyle 1+1 = 2$

- 6 years, 2 months ago

haha! this is what even i thought immediately after reading this post.. :P :P

- 6 years, 2 months ago

Lol, same here!

- 6 years, 1 month ago

Intermediate Value Theorem

If a continuous function $f$ with an interval $[a, b]$ as its domain takes values $f(a)$ and $f(b)$ at each end of the interval, then it also takes any value between $f(a)$ and $f(b)$ at some point within the interval.

Imgur

- 6 years, 2 months ago

Well, you often get to experience your teacher being Captain Obvious in your school. :P Here's something my "maths" teacher wrote on the blackboard while "teaching" us quadratics:

'If the product of two numbers is zero, one of them must be zero.'

And I was like: you don't say?

What's strange is that we always have to write this exact same line (its Bengali translation) whenever we're solving a quadratic otherwise we get a zero out of five, apparently because guys who think this is too obvious don't have any understanding of maths and just memorize stuff.

Oh, and here's another one. I found this in my eleventh grade "maths" textbook.

Law of trichotomy: Given any two real numbers $a,b,$ either $a>b$ or $a=b$ or $b>a.$

If you're thinking this isn't something important, well... I found out that there was an exercise at the end of the textbook which asked to state the law of trichotomy using examples. In fact that question appeared more than once in the eleventh grade final "maths" examination before.

- 6 years, 2 months ago

সমাধান কর

$x^2-5x+6=0$

$\Rightarrow (x-2)(x-3)=0$

দুটি সংখ্যার গুণফল শূন্য হলে এদের মধ্যে কমপক্ষে একটির মান শূন্য হবে।

অর্থাৎ,

হয়, $x-2=0$ অথবা, $x-3=0$.

সুতরাং, $x=2$ অথবা $x=3$.

নির্ণেয় সমাধান $x=2, 3$.

Alt text

- 6 years, 1 month ago

"তোমার স্টেপ জাম্প হয়েছে। $x^2 - 5x + 6 = 0 \implies x^2 - 2x - 3x + 6 = 0 \\ \implies x( x-2) - 3 (x-2) = 0 \implies (x-2)(x-3)=0$ এই স্টেপটা না দেখালে নম্বর কাটা যাবে। " -- my maths teacher

I don't want to live on this planet anymore

- 6 years, 1 month ago

এই জাম্পটা করলে ছোট ক্লাসে নাম্বার কাটা যেত। এখন আর কিছু বলে না। কারণ এই জিনিস তো আর সরাসরি পরীক্ষায় আসে না।

- 6 years, 1 month ago

Amar maths teacher sasti dito erokom step jump korle.....bolto erokom korle madhyamik porikkhai number kata jabe....,

- 6 years, 1 month ago

সব স্কুলের maths teacher গুলো অরকমই হয়। আমি RMO পেয়েছি শুনে আমার teacher বলল ওই problem গুলো নাকি S.N Deyর বইটাতে extensively cover করা আছে। -__-

- 6 years, 1 month ago

S.N.Dey te RMO r onko? Eta boddo barabari hoye gelo.....ami paper dekhechi but S.N.Dey boddo pati boi......

- 6 years, 1 month ago

jani. kintu school-er teacher guloo toh orokom. sarakkhon schooler vaater onkogulo koriye matha kharap kore chare. RMOr paper-er second questionta dekhe bole ota S.N Dey-r boi te ache, baki questiongulo ar dekheini. tokhoni bollo oi sob onko naki S.N Dey-te extensively cover kora ache, madhyamik-er age kore kono labh nei. -__-

- 6 years, 1 month ago

Amio onek loceder lecture sunechi je "madhyamik er age kore labh nei blah blah".amar obhigota bole osob lokeder kothai ekdom kan dite nei.....age t heke kora thakle onek subidha hoi

- 6 years, 1 month ago

se ekhaneo sobkota maal orokomi. INMOr age madhyamik madhyamik kore keu asol onko kortei dilo na.

- 6 years, 1 month ago

Hahahhaha.....

- 6 years, 1 month ago

(sorry, rag-er jore mukher bhasa saamlate parlam na.)

- 6 years, 1 month ago

ক্লাস ১০-এ প্রচণ্ড ভুগতে হয়েছে এই ন্যাকামির পাল্লায় পরে। একটা পাতি quadratic solve করার জন্য আধ পাতা ভাটাতে হত। ১১-এ আর quadratic solve করতে দেয় না, কিন্তু তাও দেখছি অঙ্কের থেকে ন্যাকামো বেশী।

- 6 years, 1 month ago

Amio bangali tai tomar betha bujhte perechhi

- 6 years, 1 month ago

For anyone else is wondering what @Sreejato Bhattacharya , @Eddie The Head , and I are talking about, here is a short summary:

Depressing stuff, more depressing stuff, suicidal thoughts and finally more depressing stuff.

- 6 years, 1 month ago

Haha. In a similar vein, when they teach induction in school, they are very pedantic about how the format of the proof should be written up, ranging from "state the proposition, prove the base case, prove the induction step, state the conclusion".

Staff - 6 years, 1 month ago

Yes indeed, that's very annoying. Also, whenever we're given to differentiate something, say $f(x) = x^2 + x + 1$ at $x=3,$ here's what we have to write (translated into English):

$\begin{array}{lll} f'(x) & = \dfrac{d}{dx} (x^2 + x + 1) & \quad \text{substituting } f(x) \\ & = \dfrac{d}{dx} x^2 + \dfrac{d}{dx} x + \dfrac{d}{dx} 1 & \quad \text{(the derivative of the sum of two functions is equal to the sum of the derivative of those functions)} \\ & = 2 \cdot x^{2-1} + 1 \cdot x^{1-1} + 0 & \quad (\dfrac{d}{dx} x^n = n x^{n-1} + \text{ and the derivative of the constant function is zero)} \\ & = 2x + 1 & \quad \text{(simplifying)} \\ f'(3) & = 2 \cdot 3 + 1 & \quad \text{(plugging } x=3 \text{)} \\ & = 7 \quad & \text{evaluating} \end{array}$

If we miss any of these "explanations", we lose marks. In general, the school curriculum is obsessed with jargon and nomenclature instead of mathematics itself.

- 6 years, 1 month ago

This is so very true and proves that CBSE sucks! I once lost marks for not writing that I had simplifed an expression by multiplying both sides by the same number -_-

- 6 years, 1 month ago

@Krishna Ar D'you remember the ultra-sucking method they taught to prove that the sum of all the interior angles of a triangle is 180 degrees! And you'd lose half the marks if you don't mention that "since EF is a line parallel to the base BC of the triangle, and Ab is a transversal cutting the two parallel lines, the interior opposite angles DAB and ABE will be equal."

- 6 years, 1 month ago

And if this much torture wasn't enough, there are IIT coaching institutions to go to.

- 6 years, 1 month ago

Yup,correct!

- 6 years, 1 month ago

Not only CBSE. I study in WBBSE (West Bengal state board). I often got scolded while solving a linear equation in eighth grade for not writing "transposing" when adding both sides by a certain quantity, so I know that feel.

- 6 years, 1 month ago

Truly. But nothing can beat the suckiness of CBSE. The way they literally do nothing in 9 and 10 and stuff the heck out of the child in 11 and 12 is unpardonable.

- 6 years, 1 month ago

I think you're exaggerating a little bit now.

Simplifying? Evaluating? Seroiusly?

- 6 years, 1 month ago

Yes, seriously! (মান বসিয়ে পাই... না লিখলে ১ নম্বর কাটা )

- 6 years, 1 month ago

আমি মান বসিয়ে পাই -এর কথা বলিনি [ওটা আমাদেরও লিখতে হত]। আমি বলেছি "সরলীকরণ করে", "মান নির্ণয় করে" এগুলোর কথা।

- 6 years, 1 month ago

আমাদের-ও প্রত্যেক স্টেপএর পাশে "মান বসিয়ে পাই লিখতে" হয়। ওই "মান বসিয়ে পাই"-টাকেই ট্রান্সলেট করতে গিয়ে evaluating আর simplifying হয়ে গেছে, আর কোনো ভাল ইংরেজি শব্দ মাথায় এলো না। :P

- 6 years, 1 month ago

There cannot exist an infinitely decreasing sequence of positive integers......

img

- 6 years, 2 months ago

Prove it!

- 6 years, 1 month ago

If $a,b$ be two numbers, then $a+b=b+a$.

- 6 years, 2 months ago

In $2\text{-dimensional}$ space, given a line and a point not on the line, there is exactly one line passing through that point that is parallel to the given line.

I like this fact because it is so simple and yet the source of so much controversy. See here

- 6 years, 2 months ago

If $a = b$ and $b = c$ , then $a = c$

- 6 years, 2 months ago

Given any two numbers $a,b$, either $a or $b\leq a$

- 6 years, 2 months ago

All positive odd integers can be expressed as the sum of two non-negative, consecutive integers that are less than or equal to it.

- 6 years, 2 months ago

What about $1$?

- 6 years, 2 months ago

$2k+1=k+(k+1)$

But he didn't state properly.

- 6 years, 2 months ago

Yes. What he missed is $2k+1 > k+1$ only when $k>0$. He didn't deal with the $k=0$ case properly.

- 6 years, 2 months ago

Yeah but I said "non-negative".

- 6 years, 2 months ago

I see that you have edited your comment. Initially it said that an odd positive integer can be expressed as the sum of two smaller non-negative consecutive integers, which isn't true for $1$.

- 6 years, 1 month ago

Yeah that's what I did.

- 6 years, 1 month ago

If $a=b$, then for any function $f$, we have that $f(a)=f(b).$ This same property is the one that we exploit when solving equations. (like when we add the same thing to both sides, if we multiply by the same number...)

Given any two real numbers $a,b$, there exists a real number $n$ such that $a=b+n$

- 6 years, 1 month ago

Lots of geometric properties are obviously obvious for me. For example...

• All Right Angles are congruent.
• All Straight Angles are congruent.
• Supplements of the same angle are congruent.
• Complements of the same angle are congruent.

You don't say?!

- 6 years, 1 month ago

A Line Is A Dot That Goes For A Walk!!!!!!!!!!!!

- 6 years, 1 month ago

Symmetric Property of Equality

If $a=b$, then $b=a$.

- 6 years, 1 month ago

A tangent is always perpendicular to a circle's radius.

- 6 years, 1 month ago

an odd no. added to an odd no. , gives an even number.............(1)

square of an odd no. is an odd no. ..............(2)

square of an even no. is an even no. ..............(3)

yet, there is no pythogorean triplet, in which : (odd no.)square +(another odd no.) square=(even no.) square!!!!!!!!!!!!!

- 6 years, 2 months ago

An odd square must be congruent to 1 mod 4, and an even square must be congruent to 0 mod 4. A pythagorean triplet with odd + odd = even cannot exist because 1mod4+1mod4=2mod4=/=square

- 6 years, 1 month ago

@Mursalin Habib Congratulations for completing a marvelous 200-day streak!!!!!!

Is this question an adaptation from Quora? Actually, I saw the same question, there??? There are some more questions which are good at Quora, you should add them as well.

The most obvious thing, from my perspective, are the Theorems of Euclid's Geometry and his lemma, I suppose.

- 6 years, 1 month ago

Yes.

- 6 years, 1 month ago

Factorial of 0 is same as factorial of 1. 0 as exponent is overriding (no pun there) everything else, always 1.

- 6 years, 1 month ago

Always? What about $0^0$?

- 6 years, 1 month ago

All chords are perpendicular to a circle's radius and diameter

- 6 years, 1 month ago

Any number times one is itself

- 6 years, 1 month ago

A circle's diameter is a straight line that starts from a point on the circle's edge, goes through the circle's center, and connects to another point on the edge

- 6 years, 1 month ago

A circle has one side and a 360 degrees angle

- 6 years, 1 month ago

every number added to another number gives a number greater than both of them, given that both numbers are positive and niether one of them is zero

- 6 years, 1 month ago

Infinity +infinity =infinity

- 6 years, 1 month ago

One divided by zero not exist....

- 6 years, 1 month ago

$666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2$

- 6 years, 2 months ago

The beast is coming.

- 6 years, 1 month ago

0 divided by 0 is undefined. The square root of -1 is also undefined. (Duh)

- 6 years, 1 month ago

the square root of -1 is the imaginary unit, not undefined.

- 6 years, 1 month ago

The chord of a circle meets the circle at only one point.

- 6 years, 1 month ago

That's not even true...

- 6 years, 1 month ago

'Tangent' is the word he was looking for I believe.

- 6 years, 1 month ago

- 6 years, 1 month ago

48-8/765-89412354889546/789-59+551654+46566+5*0=0 lol

- 6 years, 1 month ago

Lmfao.. i did not know a lot of these.. thank you for this note so i can start studying them hehehe

- 6 years, 1 month ago

$9^3 + 10^3 = 12^3 + 1^3$

- 6 years, 2 months ago

How is this supposed to be obvious?

- 6 years, 1 month ago

Lol

- 6 years, 1 month ago

Well, it's a really complicated branch of mathematics, taxicab numbers that is, but this is so obvious, as long as you have a calculator or a good memory. :/

- 6 years, 1 month ago

Two knot diagrams of the same knot can always be reached through a finite set of Reidemeister moves.

- 6 years, 1 month ago

Someone went to PuMaC, I see.

- 6 years, 1 month ago

The Kepler conjecture states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees have said that they are "99% certain" of the correctness of Hales' proof, so the Kepler conjecture is now very close to being accepted as a theorem. Source: Wikipedia.

- 6 years, 1 month ago

$\Huge{\color{#D61F06}{5\neq 5^0 \neq 5^{5^{5^{5^{...}}}}}}$

$\Huge{\color{#20A900}{5^{5^{5^{5^{5^{5^{... \infty}}}}}}}} \color{#3D99F6}{= 6^{6^{6^{6^{6^{6^{6^{...\infty}}}}}}}}$

- 6 years, 2 months ago