Bases - What are They? (Part 1)

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photo by lsiegert

A curious reader asked this question:

What is a base?? I’m sorry but I’m in the sixth grade and never heard of a base and then all of the sudden it’s in my homework. Will you please explain to me in easy fifth or fourth grade words what a base is? Pretend I’m stupid or something!

 Professor Homunculus replies:

Well, that’s going to be hard to pretend, because you are obviously smart enough to ask for help. You also did a good job expressing your question, so here goes:

Bases are different ways to express numbers. Like languages are different ways to express thoughts. You could say, “butterfly” in English, or “mariposa” in Spanish, “papillion,” in  French, or “schmetterling,” in German, but they would all mean the same thing, just different names for it.

You can write the number 11 in base ten, or as 21 in base five, or as A in base eleven, and they all stand for the same amount.

Just as in different languages, there are specific times you need to use different bases. That is a little hard to understand, right now, I know, but first you must learn how to translate into different bases, before you can understand anything about them.

Fortunately, it is much, much easier to learn how to translate from base to base than from language to language.

A base is the amount of digits we use to represent our numbers with.

We normally use what it called the base ten system. As you know, we normally use only ten digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make up all of our numbers. After 9, we have to start a new column (called the “tens” column, because it tell us how many tens we have). 

Then we go through the numbers again, all the way to 99, when we have to start another column, called the hundreds.

The hundreds column actually is the same as the “ten tens” (or “ten groups of tens”) column  because there are ten tens in a hundred.

Then, of course we continue, till we get to 999, when we have to represent a thousand (ten hundreds).

You see, because we only have 10 digits, and one of them is zero, we have to make a new column to carry another digit every time we get to more than 9 of anything in any column. So there is no single symbol for 10, we have to use two symbols in each of two columns.

So far, I’ll bet this all sounds simple - probably because it really is.

But what would happen, if no one had ever invented the symbols we use? Or what if they had only invented, say, two of them?

We always start with zero, because we always need a 0 to represent any empty column, no matter what base are using.

So, if you have a base 2 system, there would be only two digits, starting with 0. That means the only digits or symbols we could use, would be 0 and 1.

But how can you represent all numbers using just two digits?

It is actually simple. You will just need more columns. It usually works like this: The more digits you have, the less columns you need, and vice-versa.

In the  base 10, as the numbers grow, each higher column is 10 times the amount of the previous column. So in base ten, the columns go, 1s, 10s (because 1 times 10 is 10), 100s (because 10 times ten is 100), 1,000s (because 100 times ten is 1,000), 10,000s (because 1,000 times ten is 10,00), etc. 

In base two, it would work the same way, except each time we would multiply the column by two  (instead of ten) to get the next column. So in base two, the columns go, 1s, 2s (because 2 times one is two), fours (because two times 2 is four), 8s (because 4 times two is eight ), 16s  (because 8 times two is 16), etc.

So, in base 2,  the furthest column to the right would be the ones column (just like in our normal base 10 system). The column which is to the left of that is usually the tens column in the base ten system. But in the base 2 system it would be the twos column.

Let’s see how that would work. If we wanted to turn the number 3 (base 10) in to base two, this is how we would do it:

We would think, “Well, there is no digit for 2 or 3, so I can’t just write the 3. So I have to see if the number is high enough for there to be anything that would have to go into the twos column.”

And of course, there is. You figure out how many 2s would fit in the twos column. There is only one 2 in the number three, with one 1 left over.

So you write a 1 in the twos column (because you only have one group of 2). How many ones do you have left over? Just 1, so you write a 1 in the ones column, too. That means 3(base 10) = 11 (base two).

If that doesn’t make sense to you, read it again. If it still doesn’t, read tomorrow’s post, where we’ll continue about bases.

You may wonder why the furthest column to the right would be the ones column in all bases. There is a pretty interesting twist to the answer to that. 

You may also wonder how we represent base 1, if all we can use is one digit, and since we always start with zero, our only digit would have to be zero. If anyone knows the answer to this, please feel free to post it in a comment today. Otherwise, I’ll answer it in a future post as part of this series. 

Professor Homunculus intends to continue this series on bases until you know more about them than most college students (although, to tell you the truth, that’s not much! OK, lets say most advanced high-school or homeschool students.)

So for the next week or so, I intend to post articles about how to change from base 10 to other bases and back, how to change from non-base-ten bases to each other and back, what bases are good for, bases and their relationship to exponents, bases higher than base ten, adding, subtracting, multiplying and dividing in different bases, game theory and bases (I’ll teach you the winning strategy for a cool game), magic tricks and bases, and an awesome way to mentally add huge rows of numbers in base 10, using base 2 (it’s much easier and faster than the way we normally do it with paper, and it’s quicker than using a calculator!) 

So read on to the next post for more about bases!

Tags: base 10 , base 2 , bases , what are bases , what is a base