## Twinkle twinkle little star

Twinkle twinkle little star is a short musical piece I composed and recorded last evening.

This piece consists of 56 measures. It is 1 minute 52 seconds long. The music is composed of four tracks. This is my second attempt at recording music with multiple tracks. The last such attempt was more than two years ago when I composed and recorded 'A few notes'.

The links to the audio files, sheet music, etc. are provided below. The files reside in my website. In case, my website is down, the YouTube link provided below should still work.

The four tracks in this piece are:

1. Grand piano
2. Slow strings
4. Music box

This arrangement is based on the popular melody of the nursery rhyme called Twinkle, twinkle, little star. The melody is played with the treble notes of the piano. I wrote the bass notes for the piano and the strings, and the high notes for the pad and the music box to fill the music with emotions of love and happiness. I recorded this after about two hours of practice.

[music]

## Product of negatives

Negative multiplied by negative is positive.

This is usually taught to us in school after our knowledge about numbers expands to include the negative numbers. Most of us take this fact for granted. I took it on faith as well.

One of my pastimes is to ponder on these facts that I once learnt by faith and observable evidence rather than reason and logic. I see whether I can see these facts in a new light. A few years ago, I thought over this one as well and proving this fact was easy.

First let me provide an intuitive understanding of why it should be so. In the discussion to follow, we will assume that we already know some basic properties like distributive property, a × (−b) = −(a × b), etc. because I don't want to prove each and every thing involved here right from the Dedekind–Peano axioms.

We know that 7 × 8 = 56. Let us express 7 as (10 − 3) and 8 as (10 − 2).

So, the following must be true.

(10 − 3) × (10 − 2) = 56

Using the distributive property of multiplication over addition, we get,

10 × 10 + (−3) × 10 + 10 × (−2) + (−3) × (−2) = 56

We already know that −3 × 10 = −30 and 10 × −2 = −20. But we do not know what (−3) × (−2) is. So, let us see what we get by using what we know.

10 × 10 + (−3) × 10 + 10 × (−2) + (−3) × (−2) = 56

⇔ 100 + (−30) + (−20) + (−2) × (−3) = 56

⇔ 50 + (−2) × (−3) = 56

⇔ (−2) × (−3) = 6

We see that (10 − 3) × (10 − 2) = 56 is true if and only if (−2) × (−3) is considered as 6.

However this is not a proper proof. One may ask whether this rule is true for all such calculations even when the numbers are different. Can we prove that −a × −b = a × b is absolutely necessary for all real numbers a and b? So, let me try a proper algebraic proof.

Let a and b be two positive real numbers. We know that a − a = 0.

Multiplying both sides by −b, we get,

(a − a) × (−b) = 0 × (−b)

Again, using the distributive property of multiplication over addition, we get,

a × (−b) + (−a) × (−b) = 0.

We know that a × (−b) = −(a × b), but we do not know what (−a) × (−b) is. Using what we know, we get,

−(a × b) + (−a) * (−b) = 0.

Adding (a × b) to both sides, we get,

(−a) × (−b) = a × b