# Programming and a school algebra formula finally deciphered

### Explanation is due

I guess you also could have had the same feeling when you learnt algebra at school. Some formulas were clear and understandable, but some were cryptic and it was unclear how would anyone derive them. And then the only way to master it is to memorize it. For example, there is this known formula for a difference of squares:

a2 – b2 = (a – b) * (a + b) = a2 + ab – ba – b2 = a2 – b2 , (1)

Then, there was a little bit more cryptic formula for a difference of cubes, which is not that obvious for a regular student:

a3 – b3 = (a – b) * (a2 + ab + b2) = a3 + a2b + ab2 – ba2 – ab2 – b3 = a3 – b3, (2)

So, I think you get it and the next formula is for a4 – b4 ,

a4 – b4 = (a – b) * (a3 + a2b + b2a + b3) = a4 + a3b + a2b2 + ab3 – ba3 – a2b2 – b3a – b4 = a4 – b4 , (3)

And finally, we get to the most cryptic formula that could be frustrating in a school algebra lesson, the formula for a difference of two positive whole numbers (integers) of power of n

an – bn = (a – b) * (an−1 + an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1) , (4)

Now, the last formula seems frightening, and most interestingly one could ask, how did in the world anyone derive it? Also, how do you use it correctly?

### Take it slow

Let’s look at it in a slow motion. If we look at how we get from formula (1) to formula (4) we can notice that there is some symmetry in the numbers in the second braces in each of the formula.

So, the second braces in formula (1) have

(a + b)

the second braces in formula (2) have

(a2 + ab + b2)

the second braces in formula (3) have

(a3 + a2b + ab2 + b3)

the second braces in formula (4) have

(an−1 + an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1)

Do you see it? When there is a2 on the left side there is a corresponding b2 on the right, when there is a2b on the left, there is a corresponding b2a on the right side, etc. So this is the symmetry I am talking about. The general formula is actually a factorization of a polynomial formula. But we can look at it in a different manner, just to understand how to use it properly. The derivation of the general formula is a little bit more complex and can be found here.

One interesting thing to notice is that the sum of powers of each a, b or there multiplication ab in the second braces is always n – 1.

(a + b) = a1 + b1 , i.e. the powers are 1, 1

(a2 + ab + b2) = a2 + a1b1 + b2 , i.e. the powers are 2, 1 + 1, 2

(a3 + a2b + b2a + b3) = (a3 + a2b1 + a1b2 + b3) , i.e. the powers are 3, 2 + 1, 1 + 2, 3

(an−1+ an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1), i.e. the powers are n – 1, n – 2 + 1 = n – 1 , n – 3 + 2 = n – 1, 2 + n – 3 = n – 1, etc.

Now, also let’s pay attention that we can treat 1 as 1 = a0 or 1 = b0, and let’s look again at the expressions above

(a + b) = a1b0 + a0b1 ,

(a2 + ab + b2) = a2b0 + a1b1 + a0b2 ,

(a3 + a2b + b2a + b3) = (a3b0 + a2b1 + a1b2 + a0b3),

(an−1+ an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1)

= (an−1b0+ an−2b1 + an-3b2 + … + a2bn-3 + a1bn−2 + a0bn−1),

I hope you can see that there is a systematic pattern which is going on here.

### Rules of the game

Rule 1: The number of members in the second braces is always as the power of the initial expression, say two for a2 – b2; three for a3 – b3 etc.

Rule 2: The sum of the powers of each member in the second braces is n – 1, which was already shown in the previous examples.

Pay attention that this can be also proven by mathematical induction. But I leave it as an exercise for you.

### How to use this formula and how to zip it

Now that we’ve noticed there is a pattern this pattern show us how to use the formula in a simple way without the need in rote memorization or blindly using someone else derivation.

The only thing is to remember that the first braces always have (a – b) and in the second braces the sum of the powers of each member is n -1. Let’s look at the concrete example of a8 – b8.

Let’s start from the second braces, and write each member without powers in accordance to Rule 1. We know there should be n, i.e. 8 such members.

(ab + ab + ab + ab + ab + ab + ab + ab)

Now, let’s use the Rule 2 and add powers to each member in the second braces, remembering that for a‘s, powers start from n – 1 and decrement by 1 for each consecutive a, and for b‘s powers start from 0 power and increment by 1 for each b until n – 1. Applied to our example,

for a‘s: a7, a6, a5, a4, a3, a2, a1, a0

and b‘s: b0, b1, b2, b3, b4, b5, b6, b7

Now, putting these together in the formula we get,

a8 – b8 = (a – b) * (a7b0 + a6b1 + a5b2 + a4b3 + a3b4 + a2b5 + a1b6 + a0b7)

= (a – b) * (a7 + a6b + a5b2 + a4b3 + a3b4 + a2b5 + ab6 + b7).

### Zip it

So, now we ready to zip this formula using the math notation for the sum:

$a^n - b^n=(a - b)\sum^{n - 1}_{k = 0} a^{(n - 1) - k}{b^k}$

where k increments from 0 to n – 1, i.e. 0, 1 , 2, …, n – 1.

### An interesting turn of events

What is nice about this formula is the fact that it’s actually a concise description of an algorithm that checks whether a certain string is a palindrome.

The main idea is to take a sequence of letters (an array of characters in programming speak), and then start comparing

1. First vs. last letter
2. Second vs. one before last
3. etc
4. For each such case above check whether letters are the same. If there is at least one instance when they are not the same, then it’s not a palindrome.

In Java programming language this algorithm could be implemented as follows (run this code in online Java compiler)

public class Main {
public static void main (String[]args){
String word = "TENET";
System.out.println (isPalindrome(word));
}

static boolean isPalindrome (String word){
char[] charArray = word.toCharArray();
int n = charArray.length;

for (int k = 0; k <= n - 1; k++){
if (charArray[k] != charArray[(n - 1) - k]){
return false;
}
}
return true;
}
}


# Get Back for More Than 50 years. Thoughts on The Beatles: Get Back documentary

### Back to the… past

Have you heard about a new documentary called The Beatles: Get Back. It was created from more than 150 hours of audio and 60 hours of video while The Beatles were working on their new live performance back in January 1969. I header about this film while checking news on the BBC website. The trailer of the Get Back got me curious. The video was crisp as though it was filmed just a couple of days ago. The sound quality was good and the footage was interesting.

It turned out that the documentary was available only at Disney+ channel, so I revived my membership there just to watch this series. The Get Back consists of three episodes, each more than 2 hours long. So be ready to dedicate a good chunk of your day or even two days to be able to watch it in full, unless you intend to skim through it.

### Viewpoints

What is interesting about this series of films that you can watch it from a number of viewpoints. First, of all there is a historical viewpoint, since it was filmed in January 1969 which is almost 53 years away from now. Back in 1969, Apollo 11 landed on the Moon, Vietnam war was in its height, the Hippie movement was on the rise and solid state transistors started to make their way into electronic music equipment. So it was a hectic time to say the least. The Beatles at this period found themselves at uncertain times when the band’s future was questionable. It is also interesting to notice the fashion, cultural norms, food and other things that changed since then. For example, it is no longer acceptable to smoke cigarettes in closed public places.

At this backdrop the Get Back documentary is unwinding in its full beauty.

### Episode 1

First part of the documentary starts at the Twickenham Film Studios were the Beatles is trying to come up with more than a dozen new songs for the upcoming live performance, possibly a TV show. Some of these songs ended to be the songs for the Let It Be album. But the band finds very quickly that the place is not that good for creative process. What I find interesting, that most of the band members and their spouses smoked cigarettes inside the pavilion, threw cigarette buds on the floor which we can hardly imagine happing these days.

Looking at internal dynamics inside the band and how the creative process unfolded it was interesting to see interactions between Paul McCartney and John Lennon, some tension between McCartney and George Harrison and a calmness and wisdom of Ringo Star. What I liked to see is how the band worked together on lyrics for the songs. What I find surprising is how Ringo the drummer is able to come up with grooves on the fly without the band members telling him what to play. If we look around we see that during the rehearsal sessions musicians ate breed and butter with honey and drink tea with milk, beers and vine. What you wouldn’t notice our nowadays omnipresent mobile phones.

Speaking from the musical viewpoint, more exactly, Ringo’s drum kit I find it to be very basic 5 piece kit. It had a snare, two rack toms, one floor tom. Hi-hat, crash and ride cymbals. At the very end of episode one it can be seen that Ringo also used a splash cymbal. As for his throne he had a simple chair with a back rest. Since, apparently, there were no Drumtacks mufflers back then he used some kind of fabric that he covered the snare and the floor tom with to muffle the sound. As for the mics the kit was miced with a kick drum mic, there was also a microphone on the right side and one microphone above the kit.

### Episode 2

In the part two the musicians moved to the basement studio in the Apple building were they spent the rest of the documentary.

Turning to the Public Address (PA) equipment it is interesting to notice the vertical Fender Solid State sound speakers.

### Episode 3

I find the third part most interesting since it is in this part that the band decides it will be fun to have a live performance on the Apple Studio’s roof top. And as it turned out this decision brought with it a lot of fun for most of the people on London streets, while driving some people mad, including police officers. Also in this episode you may find how the Octopus’s Garden song came to life.

All in all, if you interested in The Beatles’ history and also want to feel how a real creative process is happening within the band during rehearsals then this documentary is mandatory for you.