In this article, I present 14 interesting math facts that I have gathered over the past few years of teaching. I generally present a collection of these to first-year students on their first day to ease them into college and also to just get them thinking. Some of these are more of an “oh, ok” level of interesting, others are a “really, how come?” level. I’ll let you be the judge. Some of these facts are self-explanatory and only take a moment’s thought, others require a little more thought and so I have made some comments along the way. I hope everyone can get something from the list.
I) The numbers on opposite sides of a die always add up to seven.
II) Zero is an even number.
For some of us, this may be a “yes, I know” fact, but for a lot of people, this is not something they have ever thought about.
Each year I propose a certain set of questions to my first-year classes to get them thinking, this is one of them as it forces them to question their definition of what an even number is. I always get the same results, everyone in the class is willing to claim they know what an even number is, but very few are willing to stand up and declare that they believe zero to be even.
For clarity, a good definition of an even number is as follows: a number is said to be even if, when divided by 2, it remains a whole number. Zero fits this perfectly since 0/2 = 0.
III) A useful trick for percentages.
Did you know that x% of y = y% of x?
This can make working out percentages a much easier task. For example, try to calculate 8% of 50 in your head. Not so easy right. Now flip it and instead work out 50% of 8, I think it’s clear which is easier.
Similarly, 32% of 75 may seem difficult to calculate, but 75% of 32 seems a much easier challenge.
IV) Every odd number, when written in English, contains an “e”.
V) “Four” is the only number, when written in English, whose spelling contains the same number of letters as the number itself.
VI) If you count up the number of letters in the 13 different kinds of playing cards (ace, two, three, four, five, six, seven, eight, nine, ten, jack, queen, king) you will find that there are 52 letters, exactly the number of playing cards in a deck (excluding jokers).
VII) The only number, when written in English, that is spelt with its letters in alphabetical order is “forty”. The only number, when written in English, that is spelt with its letters in reverse alphabetical order is “one”.
VIII) You can cut a cake into 8 equal pieces by using only 3 cuts.
I have been told by quite a few people that this has been used as an interview question by many companies to test “out-of-the-box thinking”.
The “trick” here is to not think of a cake as a two-dimensional circle, as most people tend to, but instead as the three-dimensional cylinder, which it is. This then allows us to not only make the usual vertical cuts, but we can now also make a horizontal cut. So if you use two of your cuts to form a cross on the top of the cake, effectively splitting the cake into four equal parts, and use your third cut as a horizontal cut through the centre of the cake, effectively splitting each of the four equal parts in half, you will achieve your 8 equal pieces.
IX) In a crowded room, two people will probably share a birthday.
Ok, so this is a bit vague. What does a “crowded room” mean and how probable is “probably”. Very good questions!
It turns out, and this can actually be seen very easily with some basic probability, that if you have as little as 23 people in a room there will be a 50% chance that two of them have the same birthday.
I know, this seems completely counter-intuitive. Allow me to add to that counter-intuitive feeling of yours, if the room grows to have 70 people present, you will now have a 99.9% chance that two of them have the same birthday!
This is known as the birthday paradox (or the birthday problem) and I highly suggest you look further into this. I hope to write a short article about this problem very soon.
X) There is exactly 10! seconds in 6 weeks.
For those who are unaware, for any positive integer n, n!, read as “n factorial”, represents the product of all positive integers less than or equal to n. So, for example, 5! = 5 × 4 × 3 × 2 × 1.
So, to see that 10! seconds = 6 weeks, let’s convert six weeks into seconds.
Lets now try to rewrite this so that it looks like 10!,
XI) The number of milliseconds in a day is equal to 5⁵× 4⁴ × 3³ × 2² × 1¹.
XII) Multiplying ones will always give you palindromic numbers.
For those who are unaware, a palindromic number is simply a number that is the same backwards as forwards, for example, 23432.
So, if you calculate 1 × 1, we get 1. Ok, that’s a bit of a lazy palindrome, let’s move on.
11 × 11 = 121,
111 × 111 = 12321,
1111 × 1111 = 1234321,
and keep going. If you multiply 111111111 × 111111111 you get 12345678987654321.
Furthermore, it is not necessary to have the same number of ones in the two numbers you are multiplying. For example 11 × 1111 = 12221 and
111111 × 1111 = 123444321.
XIII) 18 is the only number that is twice the sum of its digits.
Although this is easily checked to be true for 18, it does require a little bit of thinking to argue that 18 is the only number for which this is true.
XIV) The recurring decimal 0.9999. . . is exactly equal to 1.
I can give a rather simple proof of this.
Let x = 0.9999. . . .
Then, multiplying both sides of the equation by ten, we have
10x = 9.9999. . . .
If we now subtract x = 0.9999. . . from both sides we have
10x − x = (9.9999. . . ) − (0.9999. . . )
⇒ 9x = 9
⇒ x = 1.
A similar fact holds for any number containing an infinite string of 9s. For example 0.4999…. = 0.5, 19.999… = 20 and −2.999…= −3.
If I’m honest, I’m never fully happy with this proof. It certainly serves its purpose to highlight what is going on but, for those who have studied any level of real analysis, you may feel like this is a cheap trick. I somewhat agree, and for those who are interested you should look up how to prove this fact by using limits of sequences! In fact, you can see a formal proof right here
So that’s it, 14 interesting math facts to make you the life of any party, and if not, perhaps you are going to the wrong kind of parties! Thanks for reading.
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