Concepts I touch on that you should be familiar with:
- Division by 2
- Powers/exponents of 2
Have you ever wondered why our numbers have a ones place, a tens place, a hundreds place, etc.? The numbers we are used to are on a system called base ten. This means that after 10 numbers are counted in the ones place, they become a 1 in the tens place.
There are a bunch of other ways to write out numbers using different bases. In order to visualize this, we can use dots in boxes. I’ll demonstrate how we can write the number 17 in base two:
Here are 5 boxes labeled by the powers of 2 as you go right to left. The first box is 2^0 which is 1. The second box is 2^1 which is 2. The third box is 2^2 which is 4, so on and so forth. That is where we get the “base two” term from. The 1s box has 17 dots in it to start.
The only rule we have to follow with this system is wherever there are 2 dots in a box, they can be turned into 1 dot in the box to it’s left. Here is the first move:
Make sure to remove the two dots from the original box once you’ve placed a new dot in the box on the left. Continue this process until you’ve moved all the pairs into the next box. Leave any dots without a pair in their box.
The dots are thinning! Now there are only 8 dots in the 2s box. But, using our rule, we can move dots into the next box over if we take two dots and make them one:
Because there was an even number of dots in the 2s box, all of them moved over into the 4s box. We are almost done! Perhaps by now you’ve spotted a pattern: you can divide the amount of dots in the box by 2 to see how many will be in the next box over. Here we have 4 dots in the 4s box, so that must mean we will have 2 dots in the 8s box!
Now we are left with one dot in each of two boxes and cannot go any further! Now look at the box numbers and how many dots are in each. There’s one dot in the 16s box, and one dot in the 1s box. Remember how we started with 17 dots? Well 1 dot in the 16s box plus 1 dot in the 1s box equals 17! Now to write out the number we just found using base two, we will write out the number of dots in each box starting from the left. So in the first box, there is one dot; in the next 3, there are no dots; and in the last box there is a dot. So our number reads 10001. Don’t confuse this with 10,001. 10001 is its own number in the base two system.
Why use base two? Well a lot of computers use base two instead of base ten (like we use) to do computations. Because computers only have switches that can be turned either on or off, computers need a way to write numbers that only have 1s and 0s so they can either turn on or off. This is called binary (the prefix “bi-“ means 2!)
You can also use dots and boxes to turn numbers into other bases, such as base three. When you are using base three, you would label the boxes by the powers of 3 (the right box would be 1, and then 3, 9, 27, and so on) and instead of using 2 dots per 1 dot in the adjacent box, you would use three. Try writing 17 in base three using dots!
Using this model, you can see why base ten works the way it does. If you write 17 using base ten, you’ll end up with 1 dot in the 10s box and 7 dots in the 1s box. The boxes represent the “places” we are familiar with. 17 is made up of a 10 and a 7, so that makes sense.
