Sometimes there are many ideas in math that make you go

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 - http://i.imgur.com/Fvnb0Xl.png.

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TopNewestJordan Curve TheoremEvery 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 ChoiceGiven 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. – Michael Mendrin · 2 years, 11 months ago

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– Mursalin Habib · 2 years, 11 months ago

I wish I could upvote you again and again! This is gold!Log in to reply

\(\displaystyle 0\times \pi\times \tau\times \phi\times e = 0\)

\(\displaystyle\text{MIND = BLOWN}\) – Anish Puthuraya · 2 years, 11 months ago

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Reflexive property of equality\(a=a\)

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The Pigeonhole PrincipleIf 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.

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– Eddie The Head · 2 years, 11 months ago

PHP wins in my opinion :DLog in to reply

– Yan Yau Cheng · 2 years, 11 months ago

agreedLog in to reply

– Cody Johnson · 2 years, 11 months ago

agreedLog in to reply

The obvious one is

\[ 1 + 1 = 2. \]

ON page 357 of Principia Mathematica, they finally conclude

– Calvin Lin Staff · 2 years, 11 months agoLog in to reply

– Michael Mendrin · 2 years, 11 months ago

And then came along Kurt Godel.Log in to reply

Intermediate Value TheoremIf 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.

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\(\displaystyle 1+1 = 2\) – Anish Puthuraya · 2 years, 11 months ago

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– Pradeep Ch · 2 years, 11 months ago

haha! this is what even i thought immediately after reading this post.. :P :PLog in to reply

– Robert Fritz · 2 years, 11 months ago

Lol, same here!Log in to reply

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. – Sreejato Bhattacharya · 2 years, 11 months ago

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\(x^2-5x+6=0\)

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

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

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I don't want to live on this planet anymore

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– Mursalin Habib · 2 years, 11 months ago

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

– Eddie The Head · 2 years, 11 months ago

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

__- – Sreejato Bhattacharya · 2 years, 11 months agoLog in to reply

– Eddie The Head · 2 years, 11 months ago

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

- – Sreejato Bhattacharya · 2 years, 11 months ago__Log in to reply

– Eddie The Head · 2 years, 11 months 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 hoiLog in to reply

– Sreejato Bhattacharya · 2 years, 11 months ago

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

– Eddie The Head · 2 years, 11 months ago

Hahahhaha.....Log in to reply

– Sreejato Bhattacharya · 2 years, 11 months ago

(sorry, rag-er jore mukher bhasa saamlate parlam na.)Log in to reply

– Sreejato Bhattacharya · 2 years, 11 months 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।Log in to reply

– Rayeed Amir · 2 years, 11 months ago

Amio bangali tai tomar betha bujhte perechhiLog in to reply

@Sreejato Bhattacharya , @Eddie The Head , and I are talking about, here is a short summary:

For anyone else is wondering whatDepressing stuff, more depressing stuff, suicidal thoughts and finally more depressing stuff. – Mursalin Habib · 2 years, 11 months ago

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– Calvin Lin Staff · 2 years, 11 months 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".Log in to reply

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

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. – Sreejato Bhattacharya · 2 years, 11 months ago

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– Krishna Ar · 2 years, 11 months 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 -_-Log in to reply

@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." – Satvik Golechha · 2 years, 11 months ago

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– Krishna Ar · 2 years, 11 months ago

Yup,correct!Log in to reply

– Sreejato Bhattacharya · 2 years, 11 months ago

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

– Sreejato Bhattacharya · 2 years, 11 months 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.Log in to reply

– Krishna Ar · 2 years, 11 months 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.Log in to reply

Simplifying? Evaluating? Seroiusly? – Mursalin Habib · 2 years, 11 months ago

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– Sreejato Bhattacharya · 2 years, 11 months ago

Yes, seriously! (মান বসিয়ে পাই... না লিখলে ১ নম্বর কাটা )Log in to reply

– Mursalin Habib · 2 years, 11 months ago

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

– Sreejato Bhattacharya · 2 years, 11 months 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। :PLog in to reply

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

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– Ali Caglayan · 2 years, 11 months ago

Prove it!Log in to reply

Commutative Property of Addition :

If \(a,b\) be two numbers, then \(a+b=b+a\). – Prasun Biswas · 2 years, 11 months ago

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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 – Trevor B. · 2 years, 11 months ago

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Given any two numbers \(a,b\), either \(a<b\) or \(b\leq a\) – Nanayaranaraknas Vahdam · 2 years, 11 months ago

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If \( a = b \) and \( b = c \) , then \( a = c \) – Shabarish Ch · 2 years, 11 months ago

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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...) – حكيم الفيلسوف الضائع · 2 years, 11 months ago

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All positive odd integers can be expressed as the sum of two non-negative, consecutive integers that are less than or equal to it. – Finn Hulse · 2 years, 11 months ago

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– Jubayer Nirjhor · 2 years, 11 months ago

What about \(1\)?Log in to reply

\[2k+1=k+(k+1)\]

But he didn't state properly. – Mursalin Habib · 2 years, 11 months ago

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– Finn Hulse · 2 years, 11 months ago

Yeah but I said "non-negative".Log in to reply

smallernon-negative consecutive integers, which isn't true for \(1\). – Mursalin Habib · 2 years, 11 months agoLog in to reply

– Finn Hulse · 2 years, 11 months ago

Yeah that's what I did.Log in to reply

– Jubayer Nirjhor · 2 years, 11 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.Log in to reply

Symmetric Property of EqualityIf \(a=b\), then \(b=a\). – Sean Ty · 2 years, 11 months ago

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A Line Is A Dot That Goes For A Walk!!!!!!!!!!!! – Anand Raj · 2 years, 11 months ago

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

You don't say?!

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Given any two real numbers \(a,b\), there exists a real number \(n\) such that \(a=b+n\) – Nanayaranaraknas Vahdam · 2 years, 11 months ago

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A tangent is always perpendicular to a circle's radius. – Sharky Kesa · 2 years, 11 months ago

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One divided by zero not exist.... – Heder Oliveira Dias · 2 years, 11 months ago

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Infinity +infinity =infinity – Sanjeet Raria · 2 years, 11 months ago

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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 – Anand Raj · 2 years, 11 months ago

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A circle has one side and a 360 degrees angle – Nathan Blanco · 2 years, 11 months ago

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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 – Nathan Blanco · 2 years, 11 months ago

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Any number times one is itself – Nathan Blanco · 2 years, 11 months ago

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All chords are perpendicular to a circle's radius and diameter – Nathan Blanco · 2 years, 11 months ago

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Factorial of 0 is same as factorial of 1. 0 as exponent is overriding (no pun there) everything else, always 1. – Rajen Kapur · 2 years, 11 months ago

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– Cody Johnson · 2 years, 11 months ago

Always? What about \(0^0\)?Log in to reply

@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. – Kartik Sharma · 2 years, 11 months ago

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– Mursalin Habib · 2 years, 11 months ago

Yes.Log in to reply

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!!!!!!!!!!!!! – Hardik Nanavati · 2 years, 11 months ago

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– Linus Setiabrata · 2 years, 11 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=/=squareLog in to reply

Lmfao.. i did not know a lot of these.. thank you for this note so i can start studying them hehehe – Nathan Antwi · 2 years, 11 months ago

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4

8-8/765-89412354889546/789-59+551654+46566+5*0=0 lol – Samarth Badyal · 2 years, 11 months agoLog in to reply

The chord of a circle meets the circle at only one point. – Nathan Blanco · 2 years, 11 months ago

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– Cody Johnson · 2 years, 11 months ago

That's not even true...Log in to reply

– Mursalin Habib · 2 years, 11 months ago

'Tangent' is the word he was looking for I believe.Log in to reply

– Nathan Blanco · 2 years, 11 months ago

Sorry! I had meant tangent.Log in to reply

0 divided by 0 is undefined. The square root of -1 is also undefined. (Duh) – Yuxuan Seah · 2 years, 11 months ago

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– Yan Yau Cheng · 2 years, 11 months ago

the square root of -1 is the imaginary unit, not undefined.Log in to reply

\(666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2\) – Mardokay Mosazghi · 2 years, 11 months ago

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– Sharky Kesa · 2 years, 11 months ago

The beast is coming.Log in to reply

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. – Joshua Ong · 2 years, 11 months ago

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Two knot diagrams of the same knot can always be reached through a finite set of Reidemeister moves. – Cody Johnson · 2 years, 11 months ago

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– Michael Tang · 2 years, 11 months ago

Someone went to PuMaC, I see.Log in to reply

\( 9^3 + 10^3 = 12^3 + 1^3\) – Mardokay Mosazghi · 2 years, 11 months ago

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– Ishaan Singh · 2 years, 11 months ago

How is this supposed to be obvious?Log in to reply

– Sanjeet Raria · 2 years, 11 months ago

LolLog in to reply

– Finn Hulse · 2 years, 11 months 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. :/Log in to reply

\(\Huge{\color{Red}{5\neq 5^0 \neq 5^{5^{5^{5^{...}}}}}}\)

\(\Huge{\color{Green}{5^{5^{5^{5^{5^{5^{... \infty}}}}}}}} \color{Blue}{= 6^{6^{6^{6^{6^{6^{6^{...\infty}}}}}}}} \) – Aditya Raut · 2 years, 11 months ago

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