The Freshman's Dream (a classic mistake) - Numberphile

NUMBERPHILE

Video by Brady Haran and Pete McPartlan
Thanks to the Numberphile Society for error checking (special mention Francesco Fournier)

[Mikee]

Even a stopped mod 2 clock is wrong once a day.

[derphysiker]

The most common example of this mistake i see in students homework is the case for n = -1. So: 1/(a+b) = 1/a + 1/b

[guepardiez]

2ab or not 2ab, that is the question.

[Johnjkngbaup]

Hate to admit it but the square root of a sum being equal to the sum of both square roots separately is definitely something I've done like 15 steps deep into solving a differential equation on an exam before.

[Adamreir]

This was super exiting! «Here is a pattern. Does it always work: no. But it always works for primes, and here is the proof.» Great story with a great punchline, and some tension in between!

[barutjeh]

One reason the mistake is so common is that a similar rule does hold for multiplication. (ab)² = a²•b². It can make the same step feel more plausible for addition.

[md-slio]

assuming that (a^b)^c is the same as a^(b^c) is one that a lot of people make the mistake of

[adambellcomp]

Thank you numberphile!

As a composer, your video on superpermutations gave me a new way to structure elements of a composition. And now, after seeing the Pascal triangle, even though I've seen it numerous times before, has given me the answer to deriving the total number of each sub-ensemble for a given group of players.

I assumed there was an underlying mathematical answer to that problem, and I had never found the time to sit and work out an answer, so thank you!

[Steve_Stowers]

Yes, I can confirm that freshman (and even non-freshmen) make this mistake all too often.
My theory is that they do this because they've learned to think of the Distributive Law as a rule for removing parentheses, rather than a description of the relationship between addition and multiplication.

[Kaelygon]

I remember a teacher saying reducing addition from fractions was a common mistake, for example (x+2)/(3x+2) != x/3x .
So each time you did this in test, he wrote the 5 "S" reminder Senkin Suuri Sössö Supistit Summasta which from Finnish roughly translates to "you big sissy reduced from sum"

[Ryan_Thompson]

The clock thing might be nice as an example to introduce modulo arithmetic, but as a programmer, the constant switching between 1-based and 0-based systems gave me conniptions.

[EebstertheGreat]

2:55
"One o'clock, three o'clock, five o'clock"
*ROCK*
"Two o'clock, four o'clock, six o'clock"
*ROCK*
*Nine, ten, eleven o'clock, twelve o'clock, rock*
*We're gonna rock around the clock tonight*

[skylark.kraken]

In year 7 maths my teacher showed Pascal's triangle and without explaining how it was constructed asked my class what we noticed about it, I was thinking about something unrelated, nobody offered to answer, I assume my teacher looked for who was the most distracted and chose me, within a second of being asked I said that the sum of each row is equal to 2 to the power of the second number which was not an answer my teacher was expected, he asked me what else I noticed and I said that the rows with prime numbers in the second position are all divisible by that number; his reaction both times was kinda funny, he was a very animated guy. It took me until my third try to mention how it was constructed

It was beginning of the first year of secondary school and that moment was something he brought up at parents' evening, my parents bring it up whenever they can - he stood up while talking to my parents because he couldn't contain his excitement. He ended up offering me tutoring 1 or 2 lunchtimes a week, was able to get way ahead, and it all started with Pascal's triangle and prime numbers being special, he did always enjoy showing me something I had no clue about and see what I made of it. I'm a human calculator (as in a non-non-human calculator, going back to the beginning) btw

[xchurricane]

Hearing someone say "back when i was in school we called it foil, " completely agreeing with him, then realizing we're probably around the same age 😂 Don't make early career sound so old haha!

[SomeNerdOutThere]

Modular arithmetic shows up in Bresenham's line algorithm. It's a beautiful function that traces the integer values of the xy coordinates along an arbitrary line segment. Instead of using division to handle the slope, it uses modular arithmetic to tell when it should increment along the shorter axis. Because integer math is far faster on a computer than floating point, this is a very rapid way to trace a line.

[yanntal]

Fermats little theorem gives you this.
(a+b)^(p-1) = 1 (mod p)
And so we have that:
(a+b)^p = a+b (mod p) = a^p + b^p (mod p)

[faithgrins]

To answer Brady's question about "OK, but where do we use this?": modular arithmetic is _super_ important to a lot of algorithms that are used in common computing scenarios. E.g.: the most common cryptographic scheme on the internet is RSA, which only works because raising a number to the correct power in the right mod system yields the original number.

[jamestappin]

Surprised at the claim that the baseline error is made at freshman level. I remember being taught (a+b)^2 = a^2 + 2ab + b^2 at age 8.

[theexnay]

I think he undersold how useful modular arithmetic is. Anytime you just care about the remainder, it’s modular arithmetic. If you’re giving someone $4.25 in change and you want to know how many quarters to give them, it doesn’t matter that 4.25/0.25=17. All that matters is that 4.25 mod 1 = 0.25 and 0.25/0.25=1

Similarly when calculating what day of the week it is or what time of day it is, that’s modular arithmetic

Additionally modular arithmetic is great for double checking something. You can run the calculation faster in modular arithmetic and double check that matches the answer you got in normal math.

There are also a ton of ways it is used in computers and for fancy mathematical proofs. But you don’t need to know it in order to benefit from your computer using it.

[TealJosh]

For me it's just inherently ingrained from the start. Never had issue with freshman's dream. Just memorized it.
a^2{+/-}2ab+b^2 = (a{+/-}b)^2
And the true dream:
a^2-b^2 = (a-b)(a+b)