Expressing Values, and Bases

Background

Something that gets taken for granted by many people is how we write numbers. Consider - there are 10 digits we use to write numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So why is that? Can we express a certain amount of something but do it with a different number?  The answer to these questions lies within the realm of bases.

Question

Evaluate the number 50 in base 2.

Answer

50 in base 10 = 110010 in base 2

Analysis

Before we start talking about the different bases in the question, let's talk about the number 50. What does the 50 represent?

In the Roman Numeral system, symbols represented certain numbers, and using addition and subtraction rules while using those symbols would enable us to express numbers. 50, for example, is L. And counting up to and past it is:

48 = XLVIII
49 = XLIX
50 = L
51 = LI
52 = LII

and so on. Look at all the symbols it takes to express 48 - and yet I just did it here with two symbols. How can I do that?

I can do it because of the system of we all use - the base 10 system.

Let's take a look at what goes on when I express a number, say like the number 50.

There are two digits, a 5, sitting in what is called the Tens place, and a 0, sitting in the Ones or Units place. It looks like this:



When I evaluate the number, I take 5 tens (10 + 10 + 10 + 10 + 10 = 5 x 10 = 50) and add to it 0 ones (0 x 1 = 0), and so it's 50 + 0 = 50.

I can do the same thing with any number. Let's do the number 28:



That's 2 x 10 = 20 and add to it 8 x 1 = 8, so 20 + 8 = 28.

And of course we can do this with bigger numbers. Let's choose 173:



That's 1 x 100 = 100, 7 x 10 = 70, 3 x 1 = 3, and so altogether it's 100 + 70 + 3 = 173.

As I need to write bigger and bigger numbers, I can add more and more positions. There's thousands, ten thousands, hundred thousands, millions, and so on.

So where did those come from?

Notice that I can write the values of each position by using an exponential:



For the next position, the thousands position, we use . For ten thousands, . And so on. For example, the number 123,456,789 looks like this:



We can analyze any number we like using this method.

Ok - so what about using some other number than 10 as a base?

Computers, for instance, use base 2

Let's look at the number 50 in base 10 again and see what it'll look like in base 2. Let's first set up the places:



In base 2, we have only 2 symbols we can use to make numbers: 0 and 1.

Now let's look at the place holders. If I put a 1 in the 256 spot, the resulting number will be at least 256, but we want to make 50. So put's a 0 in that. Same with the 128 spot. And the 64 spot. But we can put a 1 in the 32 spot - that is smaller than 50:



Let's get those preceding 0's out of the way:



Ok, so we've taken care of 32 out of the 50 we are trying to express. We have 50 - 32 = 18 left to express.

We can put a 1 in the 16s place:



We've now expressed 50 - 32 - 16 = 2. So let's put in a 1 in the 2's place and zeros in the other places:



And so 50 in base 10 = 110010 in base 2.

Let's check it!

1 x 32 = 32
1 x 16 = 16
1 x   2 =   2
__________
           = 50

Vocabulary used:

For more information check out these links (comment to add your favourite link):

• A video from TedEd on the development of number systems and our base 10 system

Where might you have come from?

Fact-orials Index

Numbers:

• The Number 0

Operations:

• Division
• Exponentials

Where might we go?

Numbers:

• Fractions and Decimals
• Expressing Values, and Bases - Practice Problems
• Scientific Notation
• Roman Numerals

Associated Operations:

• Remainders and Modulo