Decimal System, Exponents and Perfect Numbers
Every number system has a base. It is the underlying structure the system is built upon. We’ve already seen two examples of base systems in the last section: the Babylonian sexagesimal system and the Hindu-Arabic decimal system.
The base of a number system is equal to the number of unique numeric symbols used. To determine a number system’s base ask yourself, “How many numeric symbols are available?”
Let’s begin with the decimal system.
Decimal System
The decimal system is named so because it has 10 unique symbols. These symbols are of course 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Since each place can hold only 9, we need to increase each position in our place-value system by a factor of ten.
Our first place-value is the units or ones place. We can fill this position with any numbers 0–9.
Once we get to 9, we need to start a new column to the left.
When we place numbers in this column we no longer are referring to how many single units we have. This new column represents how many tens we have. With only two columns, we can count the whole numbers up to 99.
But once we get there we’re stuck again. We now need a column to express how many hundreds we have.
Add a hundred’s column to express all the numbers up to 999.
And you guessed it, once we get there we need a column to express thousands. This process can keep going as long as we need by adding new columns to the left, each ten times larger in value than the previous.
Now we can easily dissect any number we’re given.
For example, suppose we are given the number 273. By looking at the decimal place-value system 273 has 2 hundreds, 7 tens and 3 ones.
Therefore, we can express the number 273 in expanded form as:
Now let’s get fancy by employing exponents in our decimal expansions. What’s an exponent? Glad you asked.
Exponents
Exponents are short-hand notation for expressing repeated multiplication. Let’s take a look at the most common exponent: the power 2 or the square of a number.
- What is the value of one squared?
- What is the value of two squared?
- What is the value of three squared?
See the pattern?
To square a number take the number and multiply it by itself. Why is this called squaring? True to its name we are creating a square with the given number as side lengths.
For example, I’ll use dots to represent the side lengths of common squares.
Okay, so that one takes a little imagination to see the square. How about two squared though?
And three squared:
Four, five and six squared:
… and so forth.
We’ve stumbled upon another math term: perfect square numbers. These are the set of numbers that result from squaring the natural numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … etc.
Can you guess how the exponent 3 will behave?
I’ll give you the answers to the first five numbers cubed and you figure out how I got the answers.
Think about it some. Did you get it?
The number on the right is formed by taking the number on the left, called the base, and multiplying it with itself three times.
As you may have guessed these are the beginnings of another special set of numbers called the perfect cube numbers.
If we were to make a three-dimensional cube using the base number as the side length, we would need the number of dots equivalent to the number on the right to construct the cube.
(E.g. we would need 8 dots to build a cube with side lengths of two, 27 dots to build a cube with side lengths of three, 64 dots to build a cube with side lengths of four, and so on.)
Definition of Exponents
An exponent represents the number of times the base is multiplied by itself. Here’s a diagram:
You may be wondering about non-positive exponents. Yes, they do exist (along with *gasp* fractional exponents!) but we won’t be covering those today. However, there is one simple rule I want you to memorize now.
Rule: Any number (except zero) raised to the zero power equals 1.
For example,
Back to the Decimal System
Cool, now that you’re up to speed with exponents, let’s revisit the decimal system. Fill out the decimal place-value system so that with a large number, like 70,273, it would look like this:
because
Hence, the expanded form of 70,273 is:
That’s the pro version right there!
Next Lesson: Decoding Binary
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