Base Ten vs Base Five

Let's stretch our minds a little! Let's take something that we take for granted every day in life, take it apart, mix it up, and see if it makes sense when we see it another light.

Counting! 

That is what most of us take for granted. We learn to count to 10 when we are 2 or 3 years old, and build on that for the rest of our lives. Learning to add and multiply, and work with all these numbers. Our place value depends on the number of 10, simply because of the fact that we have 10 fingers, but there are other cultures from different times that have had other bases and have learned to use numbers in completely different ways than we do, but with the same general concept. The reason we go through learning different bases is to remember how it is to learn how to think in brand new ways, just like it is for young kids who are learning math for the very first time.

Let's use this number:

1324

As we look at this, we're going to also remember our expanded notation. The 1 is not only a 1 but it represents 1,000 because it is in the thousands place. Just as the 3 represents 300 and 2 represents 20 and 4 simply represents 4. Each place value is an exponent of the number 10, which is our base. That is why it's called Base Ten. For example, 4 is 10 multiplied by ten to the zero power, 2 is multiplied by ten to the first power, 3 multiplied by ten squared, and 1 is multiplied by ten cubed. 

Now, what if we see this number and we are told it's in Base Five? That means, instead of multiplying each digit by exponents of Ten, we multiply them by exponents of Five! So it would look like this:

1324five = (1x125)+(3x25)+(2x5)+4

Notice how we are doing the same thing that we did with base 10. Except now that we are in base 5, all the numbers we multiply by are by exponents of 5. 1 times five cubed, 3 times five squared, 2 times five with no power and a simple 4. If we solve this problem we can figure out what this number translates to in Base Ten, which is our "native language" in math.

1324five = 125+75+10+4

So then, we end up with:
1324five = 214ten

A quick look at multiplying in other bases:

Here's extra help in converting to other bases