(Image from YouTube)

Video: https://www.youtube.com/watch?v=EqWZGjskZUk

…And other cool videos. So, I was watching the videos of the awesome Vi Hart, one of my favourite mathematical YouTubers, the other day because I was late for Pi Day. (The link leads to Vi’s featured video Twelve Tones, about the musical constraint of using all twelve notes in a song.) Vi’s Pi Day videos are particularly interesting because usually they aren’t about celebrating Pi but its twice-multiplied, yet often-forgotten cousin, tau – more on which you can find in her videos.

Pi Day, for those of you who were unaware, is the 14th of March – because in the American calendar the number of the month (3, because March is the third month) is put in front of the day (the 14th) so 3(month)/14(day)/2017 – whereas in European calendars it is the other way around. (so Pi Day would be the 14th March, written as 14(day)/3(month/2017(year). And Vi Hart happens to be American, as well as the creator of Pi Day, Larry Shaw in San Francisco 1998. So they use the American calendar system. Which I guess is good for Pi, because there is no 14th month. It can have a whole day to itself.

Anyway -then I wondered if there were more mathematical YouTubers – and I came across DorFuchs. Like Vi, he is a ‘mathemusician’, who combines maths and music to teach people mathematical concepts. After watching a few of DorFuchs’ videos I decided,’Tja, dieser Kerl gefallt mir’, and I went to see some more of his stuff! 🙂

One of DorFuch’s videos is a rap about multiplying two 2-digit numbers 11-19 in your head. At first I thought – ‘how?’ For example, how many people can answer this:

13 x 19 =

…in a few seconds and completely ohne Tachenrechner – without a calculator?

Don’t worry, I’ll tell you what he just said.

So in the first few seconds, DorFuchs outlines the problem: Multiplying numbers is extremely useful, and multiplying one digit numbers such as 6×7 is super-easy. But multiplying 13 x19 for instance is so hard. But, he says ‘Lernen den folgene Trick’ – learn the following Trick – and then you’ll probably be able to do it without a calculator! ‘Und dann schafft du dass veillicht sogar ohne Tachenrechner.’ (0:26)

Aber was ist das Trick?? DorFuchs singt es zu uns:

‘Nimm die erste Zahl, plus die letzte Ziffer von der zweiten.

Hang einer Null dran, und jetzt bist du bereit, denn

Wenn du das Produckt der letzten Ziffern addierst,

Dann bist du fertig und du werdest schneller wenn du das nochmal probierst.’-0:34

So I tried this out on a few numbers – of course I had to translate it first, just to check what DorFuchs was doing.

So: Nimm die erste Zahl, plus der lezten Ziffer von the imperative form of verbs. der zweiten.’ – Hmm, seems he’s using the *Befehlsform* or imperative forms of verbs – to tell us to do something. As mentioned here, the imperative is formed by taking the ‘stem’ of the verb’.

Nehemn – to take

Take the first number, plus the last digit from the second,

Hang a zero (0) next to it, and now you’re ready – then

When you add the product of the last units,

Then you’re finished – and you’ll be quicker when you try it again..

and so on.

Die Zahl – number Der Ziffer-, digit

eine Null – a zero

das Produkt – product (the result of multiplying two numbers)

multiplizieren – to multiply

addieren -to add

probieren – to try

(notice ‘mal’ can mean both ‘again’, as in ‘Noch mal’ and ‘times’ as in 18 times 19 = 342)

As DorFuchs shows us, this trick always works with any number from 11 to 19.

13x 19, I hear you ask?

13 + 9 = 22 Nimm die erste Zahl, plus die letzen Ziffer von der zwieten

22 (*10)= 220 Hang eine null dran (bzw. mal zehn) und jetzt bist bu bereit, denn

+ (3×9=27) Wenn du das Produckt der letzen Ziffern addierst

220+27 = 247

Dan bist du fertig und du werdest schneller wenn du das immer mal trainierst.

(And just in case you want to check the answer, grab your Taschenrechner and check it out.)

Online Taschenrechner/Calculator

And at 1:06 – DorFuchs explains WHY it works.

‘Doch Achtung! – redet das Trick wirklich nur von Zahlen von elf bis neunzehn an?’ -1:14

an/reden – to speak of, to address.

Does the trick really only address numbers from eleven to nineteen? How can you change the song – aka the formula – to make it work for numbers from 21-29, or 31-39 or 71-79? (That was an open question – comment away, those of you who are mathematically inclined. Meine Antwort befindet sich auf Englisch unten in den Kommentaren.)

‘Denn diesener gerade 10 plus irgendwie Ziffer von eins bis neun -okay.‘ – 1:18

Und wir rechnen ja jetzt – zehn plus a mal zehn plus b.‘ – 1:19

(10+a) * (10+b)

DorFuchs here makes the multiplication into a formula that can be true for all numbers from 11 to 19.

Any of these numbers is 10 (zehn) plus ‘irgendwie Ziffer’ – some digit from one to nine.

These unknown digits are called a und b (in English we pronounce it (Ay and Bee) – In German it’s pronounced (Ah und Beh) )

As ‘mal’ means ‘to times’ or ‘to multiply’, we are multiplying (10+a) x (10+b), so

(10 + (random number A)) mal (10+ (wahllosige Zahl B)).

1:23Das macht, wenn ich schon mal von Multiplizieren nichts übersehe,

(10*10 (+ (10*b) +(10*a) + (a*b)

DorFuchs comes to this conclusion by expanding die Klammern – like this

(10+a) *(10+b) = 10(10+b) + a(10+b)

(10(10+b) = (10*10) + (10*b) – zehn mal zehn plus zehn mal b

PLUS+

a(10 *b) = 10*a + (a*b) – zehn mal a plus a mal b )

= GLEICHT

(10+a) *(10+b) = 10(10+b) + a(10+b)

(Da die Zehne den ersten drei so man aus Klammern geht,

Konnen wir machen unseren Trick als Formula steht!)

(10+a) *(10 +b) = 10x 10 + 10*a + 10*b + a*b

(10+a+b)*10 + a*b

Nimm die erste Zahl plus die letzte Ziffer von der zweiten

Hang eine Null dran (mal 10) und jetzt bist du bereitet

Wenn du das Produkt der letzten Ziffern addierst

Dann bist du fertig und du werdest schneller wenn du das noch mal probierst!

And… damit du kannst noch mal probieren…. DorFuchs made a whole video of other examples to try on his second channel, DuFrosch – https://www.youtube.com/watch?v=CCdwcdCunD0

Danke schӧn DorFuchs, and I’ll definitely check out some more of your mathematical raps sometime!

Vocab:

Imperative forms: Because he’s telling us directly to do something, DorFuchs uses the imperative with verbs such as ‘Nimm’ and ‘Hang’ The imperative of ‘nehmen is’nimm’ or ‘hangen’ to ‘Hang’.

Mathematical vocab: addieren – to add or – ‘plus’ +: but pronounced ‘P l UU s’ rather than English ‘p l uh ss’

multiplizieren – to multiply, or ‘mal’, to ‘times’, as in 6 times 7 = 42. Notice also ‘mal’ is used for times in other ways, such as ‘noch mal’ -I like to think of it as ‘still (another) time’

Die Zahl-(plural Die Zahlen) – number(s)

(And, just in case, eins(1), zwei(2), drei (3), vier(4), fuenf (5), sechs(6), sieben(7), acht(8), neun(9), zehn (10), elf, zwolf, dreizehn (13), vierzehn (14)… und so weiter…)

Der Ziffer – digit

(What does this have to do with PI Day? Check this out by DorFuchs- Pi ist irrational!

(I still don’t know why.)