A Quick Intro to Number Bases

by rrusczyk, Jun 17, 2006, 3:21 PM

This is the second in the series of articles I'm writing in preparation for a conference this summer. This story is intended for gifted elementary school students and middle school student who haven't yet mastered number bases. Parents of younger students may want to work alongside their children with the story, especially the extensions at the end. The story covers material extremely quickly!

What If We Didn't Have Ten Fingers. . . But We Still Had 10 Fingers!

When I was 14 years old, I was part of a special government experiment. They had built a fancy new doo-dad that could send me into alternate universes in which everything was changed, but only slightly. They invited me to be part of the experiment because I was very good at math. They needed someone who was good at math because they couldn't understand the math in these new universes. They knew I was good at math because I was very fast at finding the final digits in addition problems, subtraction problems, and multiplication problems. AND I DIDN'T EVEN USE A CALCULATOR!

Here are a few of the simple computations in which I can find the final digit very quickly. By final digit, I mean the units digit.

4 + 6
15 + 13
23 - 9
6 x 4
7 x 3

Yeah, I know, they're pretty simple. Click here for my answers.

But I can also do complicated ones, too! Like these:

534 + 9325
9103 - 1902
45 x 72
45539 + 4391
54820 - 39542
574937 x 43423

Can you do those quickly? I can. Click here to see what I get

Did you get those answers? How long did it take you? It only took me about 7 seconds. Like I said, I'm fast. Click here for a little hint how I do it so fast.

Can you find a rule to use that will let you find the last digit quickly for the following:

21 x 34
31 + 54
98 - 17

Click here for a hint.

Do you have the rule now? Click here to see how I do it!

So, like I said, I was pretty good at doing these last number thingys, so they figured I must be good at math. You can do the last number thingys now, too, so you must be pretty good at math now, too. Maybe next time, it will be your turn to do the experiment.

Which brings me back to the experiment. In the first experiment, I was sent to a universe in which everyone had only one hand. Even weirder, their hands had seven fingers! Even weirder than that, their math had all sorts of things like 3 + 6 = 12, which is very weird because everyone knows that 3 + 6 = 9. And yet, all their math-stuff seemed to work. Their airplanes still flew, their computers still worked, and their clocks, well, I guess they worked, too, but I couldnt understand them at all at first.

I was pretty confused when I first started looking at their math. So, I did what I always do when I get confused. I went back to the basics. It's tough to get more basic than counting, so I asked them to count things for me. They thought this was pretty funny - some weird kid with two hands that only had 5 fingers each asking them to count things. I got pretty upset, and said that I didn't like them laughing at me because I had ten fingers.

Then they said, 'Ten? What is ten?'
I said, 'It's the number after nine!'
They said, 'Nine? What's nine?'

You know what happened next. They didn't know eight, either. They did know seven, though.

When I said seven, they held up all their fingers and said 'Seven!'. I was relieved, and wrote a 7 down.

They pointed at it and said, 'What's that?'
I said, 'It's a 7!'
They said, 'No it isn't!
This is seven -' And they wrote. . . . 10!

Can you believe it?!? 'That's ten!' I screamed.
'What's 'ten'?' they said.

Now I was really confused. So, I asked them to count for me and to write down the numbers as they counted. So they said, 'One, two, three, four, five, six, seven, eleven, twelve, thirven, fourven, fifven -' And if you think that's weird, what they wrote was even weirder!

They wrote '1,2,3,4,5,6,10,11,12,13,14,15'. No 7, no 8, no 9!!! I asked them to count again, but this time I watched their fingers, because they liked to count on their fingers (I admit, sometimes I do too

). When they had a full hand-full, that was seven, which they wrote as 10, then they kept counting. When they got to two hand-fulls, they would write that as 20, and so on.

I realized that this is the same thing I do! When I use all of my fingers, I write 10. Then I keep counting. When I use all my fingers twice, I have 20. But they have less fingers than I do! So they get to 20 before I do! I got it! When they would fill up their fingers seven times, they would write 100, just like when I fill my ten fingers ten times, I write 100. But, it just takes them less fingers to get there.

So, I figured out how to turn their numbers into my numbers. I didn't want to mix the numbers up, so I put a little 7 after their numbers. They thought this was pretty funny. But it let me make some sense of their numbers. I counted with their numbers and my numbers right next to each other:

$\begin{eqnarray*} \text{Their Numbers} && \text{My Numbers}\\ 1_7 &=& 1\\ 2_7 &=& 2\\ 3_7 &=& 3\\ 4_7 &=& 4\\ 5_7 &=& 5\\ 6_7 &=& 6\\ 10_7 &=& 7\\ 11_7 &=& 8\\ 12_7 &=& 9\\ 13_7 &=& 10\\ 14_7 &=& 11\\ 15_7 &=& 12 \end{eqnarray*}$

And then I started to see how to turn their numbers into ours. See if you can do it, too. Keep counting with their numbers (pretend you only have seven fingers if you have too).

Click here to see what I get when I keep counting with their numbers

I could see that in their numbers, their units digit was just a number of fingers, just like ours. But their next digit over was a number of sevens, not a number of tens! So, $23_7$ means 2 sevens and 3 ones, for a total (in our numbers) of $2\times 7 + 3 = 17$ in our numbers.

But what happens when they add 1 to $66_7$ ?

Then, they have seven units, so that makes a bundle of seven. But then they have seven bundles of seven, so . . . They need a new digit, just like when we add 99 + 1! So, $66_7 + 1_7 = 100_7$ . The digit we used to think of as 'hundreds', or $10\times 10$ , to them is the number of bundles that have seven groups of seven, or $7\times 7$ . And so on for the rest of the digits. Then it made sense to me. We write our numbers like 4319 when we really mean

$4316 = 4000+300+10+6 = 4\times 10\times 10\times 10 + 3\times 10\times 10 + 1\times 10 + 6\times 1.$

But when they write $4316_7$ , they mean
$4\times 7\times 7\times 7 + 3\times 7\times 7 + 1\times 7 + 6\times 1.$

Using this, I could turn their numbers into ours pretty easily. See if you can find what these numbers are in our number system:

$45_7$
$51_7$
$100_7$
$203_7$
$4130_7$

Click here to see what I think they are.

My adventure continued. . . Here are some of the things I had to think about:

(Note - most of these will be very hard to figure out the first time you try. Be patient, and you might want to ask your parents or older students for some guidance! Also, you might like the interactive Base Blocks tools you can find by clicking here as a way to play with different number systems. Those tools were developed at Utah State University through an NSF grant.)

1) Once I could figure out what their numbers were, I could figure out how to add them. At first, I just converted their numbers to ours to add them, but then I realized I'd have to convert back from mine to theirs. That seemed like a pain, so I thought about if I could add without converting to my numbers. How did I add their numbers without converting to mine first?

2) Of course, they became curious about my numbers, so they wanted to learn how to convert my numbers into theirs. It took me a while, but finally I learned how to do it. How did I do it?

3) One thing I noticed was that they could do simple multiplication faster than I could. Soon I realized why - they had a smaller multiplication table to memorize! They only had to go up to $6\times 6$ , and didn't have to deal with things like $8\times 9$ . Can you make their multiplication table (from $0\times 0$ up to $6\times 6$ )?

4) They did have to do a little more addition when they multiplied, though. Can you use their multiplication table to multiply $43_7\times 31_7$ without converting to our numbers first?

5) Finally, I told them about how I could find last digits very quickly. They said that they could do it, too! See if you can figure out the final digits of each of these (in their number system, not ours):

$41_7 + 65_7$
$435_7 + 541_7$
$6103_7 + 54_7$
$46_7 - 33_7$
$531_7 - 52_7$
$12_7 \times 21_7$
$412_7 \times 6_7$
$461_7\times 33_7$

6) I finally figured that out, and showed off a little by saying I could even figure out the last digit in their number system of $(3_7)^{63}$ . How did I do it? (Hint: Try doing it in our number system first!)

7) After I came back from this experiment, the scientists who sent me were so happy they sent me on another one. In the next universe, the people only had four fingers. I felt pretty confident I would be able to understand their math, but then I saw it and it looked like this: $\alpha + \beta = \gamma$ , $\alpha + \gamma = \alpha\phi$ , $\beta + \beta = \alpha\phi$ , $\alpha + \phi = \alpha$ , and the one that really got me confused, $\gamma\phi\alpha - \beta\phi = \beta\beta\alpha$ . What in the world was going on?!?