Famous mathematician puzzled by child's homework

hey this is press tow Walker here's a problem that was posted on Twitter recently and generated a lot of discussion the problem comes from Timothy goers who is a mathematician at Kad CH France and the University of Cambridge he was also awarded the fields medal in 1998 the fields medal is considered the Nobel Prize for a mathematician so he posts here's a Puzzler for my daughter's Math's homework how many degrees inside the following shapes triangle rhombus Circle and Pentagon so to many people this may not even seem like a controversial question so the sum of the interior angles in a triangle is 180° the sum inside of a rhombus which is a four-sided quadrilateral is 360° then let's skip ahead to Pentagon which will be 5 140° but then we get to the question of the circle your natural instinct might be to write 360° because they're 360° In A Circle however if you think about the question as it's stated which is to find out the number of degrees inside the following shapes you might be thinking this question is referring to the sum of the interior angles of a circle in which case you might think the answer is infinity in fact this is what his daughter put down and what many mathematicians were saying should be the correct answer so to understand where this answer of infinity comes from let's take a step back the measure of a straight line is 180° what is the sum of the interior angles of a triangle we will solve this problem with a clever construction that's been known since ancient times construct a line that goes through the bottom side of the triangle according to the postulates of ukian plane geometry we can then construct a unique parallel line that goes through the top point of this triangle we now have alternate interior angles on both sides of the top angle so the left angle of this triangle on the bottom is equal to this angle on the top the same thing is true the bottom right angle is equal to this angle on the top we can now see that the three angles of the triangles have exactly a sum that's equal to that of a straight line therefore the three angles sum up to 180° notice this is true no matter how we draw this [Music] triangle we will always have the sum of the three inter angles is equal to [Music] 180° we can then build from this to figure out the formula for a general polygon so we have a triangle will have a sum of interior angles equal to 180° now what about a quadrilateral the clever trick is to divide this with two opposite vertices we now have divided this quadrilateral into two different triangles this triang triangle will have a sum of its interior angles equal to 180° and the same is true for this triangle so the total sum of the interior angles of this quadrilateral will be the same as two triangles which is 180 + 180 which equals 360° if we go to a five-sided Pentagon we can divide it up into three different triangles and each of the triangles will have a measure of 180° so the total interior angle sum in a pentagon will be equal to 540° so when n is equal to 3 for the number of sides the interior angle sum is 180 when n is equal to 4 we have 360 when n is equal to 5 we have 540° so each time we increase n we need to add 180° so we can easily come up with a formula for a general n ided polygon and this will work out to be 180° multiplied by the quantity nus 2 so if we take this idea we start out with a polygon that has an interior angle sum of 180° multiplied nus 2 we can see how this will keep increasing as the number of sides n keeps increasing but as we take the limit as n goes to Infinity this polygon is going to approach the shape of a a circle so it would suggest that the sum of the degrees inside the circle has to be a number that keeps growing infinitely large so this is the justification that the answer is infinity so now let's return to the homework question Timothy G's daughter did put Infinity as a number of degrees inside the circle he then posted an update a few days later update the mystery is now solved the answer to the question is 360 my daughter put Infinity which was marked wrong so she got 39 out of 40 on the homework but that is how you learn many people were upset with the grading they felt that either Infinity should have been accepted as the correct answer or the question could have been thrown out because it was poorly phrased but I want to offer one more perspective this whole exercise got me thinking why do we say the number of degrees in a circle is 360° and does this question actually make any sense so let's go back to a basic fact we started out with our derivation by saying a line measures 180° I could trace this back to uclid who said that a line that is divided exactly in half is equal to two right angles which nowadays we would say is 90+ 90 which which is 180° however why is this 180° why is this given as a fundamental measure the answer traces back to a circle ancient societies including the Sumerians Babylonians Egyptians Indians Chinese and Mayans were studying astronomical features and noticed the year was about 365 days they noticed a circular pattern about this and it was convenient to round this down to 360 and divide each day into a degree so the circle became divided into 360 days which is 360° it made sense to measure the angle of The Arc in terms of this central angle so one full turn around the circle was 360° and you could subdivide either this Arc or you could divide the central angle by 360° so imagine you go halfway around the circle this form a straight line across the diameter the measure of the central angle would have to be exactly half of 360° which is 180° so to get the chronology right a circle is divided into 360° this is why it makes sense for a line to be 180° then a triangle has the same interior angle sum as a line which becomes 180° and all of the formulas for the polygons follow on this basis so it is the circle of 360 Degrees that's the fundamental measure so we now go back to the homework question and think about the phrasing people are thinking about what is the meaning of degrees inside well for the triangle and the rhombus and the Pentagon inside is referring to the sum of interior angles but in a circle it would refer to the measure of its central angle so you would say you're using one word to mean two different things which should never happen in mathematical homework this is a poorly phrased question it should never be allowed but I want to offer another perspective think about the sentence there is no time like the present to present a friend with a present we have one instance where present refers to a time another instance where it refers to an action of giving and a final instance of the word where you're actually giving a physical or otherwise emotional object so we have a single word that has three different meanings in the same sentence I would say many of you should understand this question without any difficulty it's not a problem if you're using the same word to have multiple meanings even in the same sentence that's just a part of life now since we're talking about language I POS question to AI so Gemini I said how many degrees are inside a circle and the response was there are 360° inside a circle this is a fundamental measurement in geometry in fact Gemini got tripped up with the other question how many degrees are inside a triangle the response was a triangle itself doesn't hold degrees inside it however there's an interesting fact about the angles of a triangle the sum of the degrees of all interior angles in any triangle angle will always be 180° so if you look at Gemini which is based on a large language model it was actually able to parse out that the degrees inside a circle are different than the degrees inside a triangle and it was able to come up with the right answer I pose the same question to chat GPT how many degrees are inside of a triangle the total number of degrees inside a triangle is always 180° this property is known as the angles sum property of triangles no matter the size or shape of a triangle the sum of its interior angles will always be equal to 180° I then pose the question how many degrees are inside a circle a circle does not have interior angles in the same way that polygons do instead a circle is made up of infinitely many points and any angle formed inside a circle would be measured in radians rather than degrees however if you're referring to the central angle of the circle it's measured in degrees and the central angle of an entire circle is 360 degrees so for me I think it is fair game to ask how many degrees are inside a polygon or inside a circle although degrees inside would have different meanings in those different contexts but I pose a question to you do you think this question was fair and what answers would you have given thanks for making us one of the best communities on YouTube see you next episode of mind your decisions where we solve The World's problems one video at a time oh

Channel: MindYourDecisions

Views: 121,235

Keywords: mathematics, math, maths, riddle, brain teaser, puzzle, math puzzle

Id: eA8THVkRXXw

Length: 11min 40sec (700 seconds)

Published: Sat Jun 01 2024