Math Problems

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  Handouts by Prof. Loh
  • tidyboar  

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  @tidyboar
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  Math question help 2.0
  • The Blade Dancer  

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  @The-Blade-Dancer The factor of \(2^{19}\) comes from all the even numbers in the factorial \(N!,\) like \(2, 4, 6, 8, 12, 14, \ldots\) etc., in addition to extra \(2\)'s that come from the powers of \(2\) like \(4, 8, 16.\) Actually, though, you know that \(N\) must equal \(22,\) since we have two \(11\)'s (from the fact that \(11^2\) is a factor). One is from \(11,\) and the other is from \(22.\) We know that \(N\) can't be larger than \(22,\) so \(\boxed{N=22}.\) And then \(N!\) (N factorial) would be \(22! = 1 \times \textcolor{red}{2} \times 3 \times \textcolor{red}{4} \times 5 \times \textcolor{red}{6} \times \textcolor{blue}{7} \times \textcolor{red}{8} \times 9 \times \textcolor{red}{10} \times \textcolor{yellow}{11} \times \textcolor{red}{12} \times 13 \times \textcolor{purple}{14} \times 15 \times \textcolor{red}{16} \times 17 \times \textcolor{red}{18} \times 19 \times \textcolor{red}{20} \times \textcolor{blue}{21} \times \textcolor{orange}{22}.\) I tried to color-code so that all of the \(\textcolor{red}{\text{red}}\) numbers are even and contribute factors of \(2\) (this is where you get the \(2^{19}\)), the \(\textcolor{blue}{\text{blue}}\) numbers contribute factors of \(7\), and the yellow numbers contribute factors of \(11.\) Since \(\textcolor{purple}{14}\) is a multiple of both \(2\) and \(7,\) I made it purple. And since \(22\) is a multiple of both \(11\) and \(2,\) I made it orange.
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  Math question help
  • authenticwasp  

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  @authenticwasp How many digits can the number have?
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  Interesting math problem
  • tidyboar  

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  Yeah I also found a=3 and b=2. I'm pretty sure there's no more solutions but I don't know how to prove that
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  Pick’s Theorem
  • RZ923  

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  @RZ923 That's a very good question! I never learned the proof, but I believe it involves induction. You can more easily prove that Pick's Theorem works for simple shapes like triangles and rectangles. Using these smaller shapes, you create any polygon whose vertices lie on lattice points (points with integer coordinates). Sorry that I don't know the proof right now, but I'll try to look into it someday!
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  “You cannot divide by zero!” - but why...?
  • nastya  

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  Try using a tablet, i guess?
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  1 to the 0 power
  • The Blade Dancer  

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  lol you have a point there
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  About 0/0
  • RZ923  

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  Hi @RZ923! Well, that's a nice question and is a very deep concept. You can read about it here.
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  I came across some problems and want to know what you think.
  • smartlion  

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  @smartlion You can also bring these questions to the Ask Math Anything live stream and ask Prof. Loh! That is probably the best way, since we (the moderators) are often too busy with other work to answer non-course-related math questions.
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  Is there a way to prove Euler’s Identity without using trigonometry?
  • RZ923  

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  @nastya Thanks
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  Volume of a sphere
  • RZ923  

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  @nastya thanks
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  Who invented the strategy for converting repeating decimals to fractions?
  • The Blade Dancer  

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  Hi @The-Darkin-Blade! The method of converting repeating decimals to fractions is just one of the algebraic methods of working with rational fractions. It is just an algorithm that tells us what to do. It is the same as the method of solving equations, inequalities or changing fractions to a reduced form. Honestly, there is not only one algorithm of converting repeating decimals to fractions - there are lots of them. Some of them are harder and require some background, some of them are easier and do not require any prior knowledge. So, once again, it's just an algorithm. For example, mathematicians use the method of moving a decimal point not just for converting repeating decimals to fractions, but also in many other cases.
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  Explain please Area of circle
  • The Blade Dancer  

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  Hi @The-Darkin-Blade! Nice to hear from you again! The area of circle is a veeery big topic for a small forum post. There are lots of articles about the formula of a circle's area and other things related to this topic! Hundreds! Even thousands! Your confusion is normal To learn more about it, you can start from here, or find more advanced information here. You can also imagine that a circle is a regular polygon with lots and lots of vertices (actually, infinitely many vertices). To find the area of a regular polygon, you cut it into isosceles triangles (see picture) and then sum up their areas. So the area of a polygon is equal to \(n\times(\frac{1}{2}ah)=\frac{h}{2}\times(na),\) where \(n\) - number of vertices, \(a\) - side length and \(h\) - distance from the center of polygon to its side. The term \(na\) is the perimeter of the polygon. As the polygon becomes more and more like a circle, this value approaches the value of the circle's circumference, which is \(2\pi r,\) and value of \(h\) approaches the circle radius. So, substituting \(2\pi r\) instead of \(na,\) we get: \(\text{area of circle}=\frac{r}{2}\times(2\pi r)=\pi r^2.\) Studying this in more detail and discovering lots of new and very interesting things will be easier as you get older and study higher-level topics like functions, derivatives, integrals, and many other things.
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  another question
  • Kevin Li  

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  @Kevin-Li We will get back to you with this... we're thinking about it!
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  Discussion question
  • Kevin Li  

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  @Kevin-Li You have the right idea! The key is that the only time the amount of money changes is when Boris trades in a penny for 5 quarters... But, does this always mean that the amount of money is multiplied by 125? Here's an example: If I have two pennies and a nickel, that's 7 cents. If I trade in one penny for 5 quarters, i'd have one penny, one nickel, and five quarters, and that's 131 cents. Hm, my money didn't exactly get multiplied by 125. What exactly changed? Once you figure that out, you'll get the answer in no time!
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  Euler
  • The Blade Dancer  

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  @TSS-Graviser It's \(-1\) because the point in question is point \(A,\) which is at \((-1,0).\) Why does \(e^{\pi i}\) give us this point? Because \(\pi\) corresponds to the angle \(180^{\circ}\) in the picture!