Gelfand: Chunk 5 - Powers: Positive and Negative

December 10, 2016

For me, powers always made sense, but yet again it turns out that I’d missed some subtleties which are now evident as I come back around using the Oakley chunking-cum-visual-grokking approach.

General Definition of Powers

Let’s kick off with the lovely General Definition.

For positive integers ​,

(​ times)

There’s something worth noting explicitly about this general definition, and it’s a concept we’ve not come across in this series; the presence of a mathematical “convention”.

The Convention of Negative Powers and Fractions

So what is the convention? Well, it turns out that when we write ​ as ​ we’re relying on a convention. But a convenient one.

This convention helps because it unifies how we conceptually treat very large and very small numbers.

But why the “convention” piece? Because it is a convention rather than a fact it is impossible to prove that ​ because ​ makes no sense without the “​” agreement.

Interesting huh? Let’s continue.

Tricks Arising

With this in place, we can move on to the fun, useful stuff; the tricks we can slot into our memory banks.

Shorten Things

The long expression

can be written in the shorter form

and you can also write

in the form

Negative Powers to Fractions of Powers (& Back)

As it said in the definition,

This means we can take advantage of this equality to re-phrase things in either of the formats.

E.g. re-writing to use only positive powers

and

And don’t forget you can go the other way too.

Everything has a Power

If a number is just a number, and seems to have no power, then it secretly has a power of ​

This even works when the base is zero

In fact, when the base is zero, the power can be anything

etc.

Multiplying Powers

When multiplying powers with the same base, just add the exponents and keep the base the same:

This works for negative powers too:

That example had only one negative power, but more, even all can be negative and the rule still holds.

And it also works when one or more exponents equals zero

In general

Use Powers to Multiply Small and Big Numbers Conveniently

You can lean on powers to allow you to multiply small and big numbers with ease (and without a calculator).

E.g., to multiply ​ and ​

If Powers are Different, Make the Bases the Same

In Gelfand Problem 55(b), we have ​ on one side and ​ on the other. To get to work on this, both sides need to be the same power. Therefore

Later on, in Problem 55(f), we have ​ on one side and ​ on the other. Again, we need the same power.

If Powers are the Same, You Can Add the Bases

In Gelfand Problem 55(g), we have ​. The bases are different, but the powers are the same, so

Pay Attention to the Question

Problem 49 asks for “the nuber of digits” rather than “the answer”. Gelfand gave us a clue just before as to how to represent ​ in terms of ​ (it’s ​). Thus, when we have to raise ​ to ​ (or ​) we can move across to powers of ​ and only care about the bit which will have the most digits: ​. We can stop worrying about ​ as it will have less.

(The answer is ​ by the way, a ​ followed by thirty ​s.)

Powers of 2 are Quite Close to Powers of 10

(Close enough for computer scientists anyway.)

In Gelfand’s “Remark” on page 26, we saw that ​ (​) is pretty close to ​ (​), close enough for computer scientists to be happy enough to ignore the ​ extra bytes in order to be able to use the term “kilobyte”.

This then brought up the fact that other powers of ​ were not a million miles away from some powers of ​, and that when we were calculating numbers of digits rather than the actual result, then this was useful.

Break Out Powers to Simplify Things

Last in this post is this handy trick which builds on some of the chunks we identified for simplifying things in post 2 (Commutative, Associative and Distributive Laws)

E.g.

Terminology

Gelfand: Chunk 5 - Powers: Positive and Negative - December 10, 2016 - {"name"=>"Andrew Harmel-Law", "github"=>"andrewharmellaw", "twitter"=>"al94781"}