Shifting and Zero Powers

9 y.o. Daize and I stumbled onto another math thing (I mentioned it briefly a few days back), and I’ve been aching to write about it so here goes.

Daize and I started talking about binary numbers years ago.  At the time Daize wanted to build her own phone, so we talked about how computers work—in binary.  From there, things just kind of kept rolling becasue everything is easier in binary for us.  When you want to talk about addition, of multi-digit numbers for example, and how to carry and all that good stuff, it’s simple to talk about binary where there are only two numbers you need to know: zero and one.  With memorization out of the way, Daize and I could talk about carrying and adding to our hearts’ delight.  Same deal for multiplication which then led us to raising numbers to powers, and here we are!

Raising numbers to powers—-for those who don’t mess with this stuff every day—iis usually illustrated by multiplying numbers by themselves.  So, 2 to the 2nd power is just 2 times 2, and equals four.  Two to the third power?  2 times 2 times 2 which equals 8, and so on.  Five to the second power is 5 times 5 equals twenty-five.  You get the idea (if not, head for the comments, and we’ll talk).

As a kid in math class, the explanation for two to the first power bugged me.  The answer is two, but what are we multiplying together to get there?  I don’t know. Two to the zeroeth power was even worse.  The answer is one.  5 to the zeroeth power?  One.  117 to the zeroeth power?  You guessed it!  Also one.

Teachers told me it was a definition, and to just accept it, and so on we went.  Any number to the first power is itself.  Any number to the zeroeth power is one.  Heavy sigh.

But thankfully, last weekend, talking about math with Daize, the beauty of number bases finally showed me the way!

First, number bases.  They’re weird until you get used to them because we use a single number base for just about everything.  I’ve written an entire post on bases, but here are the basics.  We’re all used to base ten where there are ten different ‘numbers’ we can use, 0 - 9.  If we want to write down a ten, we do it by placing a 1 in the ten’s place.  Binary is a base most people have heard of, it’s base two.  That means we only get two ‘numbers’ (characters?) to play with, zero and one.  So, when we want to write down 2 we do it like this: 10.  It’s just that now that we’re in base 2, that 1 is in the two’s place.  The number furthest to the right?  That’s always the one’s place.  One more example, what if we had base five?  Then we’d have 0 - 4 to play with, five would be written as 10, and seven would be written as 12, (one in the five’s place, plus two in the one’s place equals 7.)

Changing bases doesn’t change a number, it just changes how we write it down, so seven is 7 in base ten, 12 in base 5, and 111 in base 2, (that’s a four plus a two plus a 1).  So, we’re leaving the number the same, but changing how we represent it.  And therein lies the chance to see new things.

Playing with binary multiplication, Daize and I saw pretty quickly that multiplication is just shifting the number one to the left.
1 x 10 = 10 (2 in base ten)
10 x 10 = 100 (4 in base ten)
100 x 10 = 1000 (8 in base ten)

So, every time we multiplied by the base, (2), we just had to shift the 1 to the left one place. 

Alternatively, every time we shifted 1 to the left by one digit, we’d multiplied by the base (2).  That led to other cool discoveries like in its own base every number raised to the second power, (squared) is 100. All of tghis is stuff that people already knew, but it was new to us, so it was fun.

Anyway, with base 2, Daize and I had a way to talk about addition and multiplication in more simple terms.  Life was good.  We didn’t have to memorize big tables of numbers—either for addition or multiplication—if we didn’t want to.

But this number base thing?  It’s just the gift that keeps on giving, because watch this.

Suppose I don’t shift the number one in a given base.  So, I shifted it zero times right?  In programming we use ‘<<’ to signify shifting a number to the left, so:

1 << 0 = 1

Now, suppose I shift it to the left by one place:

1 << 1 = 10 (which would be ten in base 10 or two in base 2 or five in base five.)

And if I shift it by two places?

1 << 2 = 100 (which would be one hundred in base 10, twenty-five in base 5, and four in base 2.)

The point is that while it’s nice we talk about multiplying when we talk about raising a number to a power, what we’re really talking about is shifting the number one in that number’s base to the left.  Now the zero and first powers make sense.  Don’t shift at all, (the zeroeth power)?  The answer is always 1.  Shift one by one in that base?  The result will always be the same as the name of the base. After you get up to the second power, yeah, the multiplication thing works, but with shifting, you can get all the way to any answer and you don’t have to have definitions!

But there are fancy words
Our 1 to a specific base is called a ‘generator’ under exponentiation (raising numbers to a power), for its specific base.  Give me the number 1, tell me the base it’s in, and I can shift it to the left to generate every power of the base.  So, I can get all the answers just by starting with one, (the generator), and performing the operation, (shifting), as many times as I need to.

Where does that leave us with logarithms?
I don’t know.  You might remember from grade school that if I raise a number to a power, that’s called exponentiation, and if I want to do the inverse of that I take a logarithm of the answer.  So, when I exponentiate 10  to the second power, (shift one to the left by 2), I get 100, and when I take the logarithm of 100 (in base 10), I get back 2.  What cool shifty/basey things will the gang and I find out about logarithms?  I’ll keep you posted. 

In the meantime, what are your favorite math tricks?  Also, did any of this make any sense, or am I just tripping on public school math memories?