Numbers and Scientific Notation

This will look pretty familiar if you’ve recently taken a math course. If you last touched math in high school, it should still look familiar to you. We are going over this to make sure we are well grounded in mathematical techniques and symbolic reasoning before going into the core material. If it seems overly easy, that’s probably a good thing.

Number Line

Rational numbers are listed on the number line using interval notation. Interval notation uses two symbols to explain what kind of interval it is. The first, using square brackets, represents a closed interval. An example is [0, 10], which includes the numbers 0 and 10, and all the numbers in between, such as 4. The second form, represented by parentheses represents an open interval. The example (0, 10) includes all the numbers between 0 and 10, but not 0 nor 10 themselves. So 4 is included and so is 0.0000000001. But not 0. It is also possible that one side is open and one closed.

Also important is understanding one side may go all the way to infinity. This is represented with $\infty$, but this is only used with parenthesis, by tradition. By convention, it is considered illogical to have a closed interval at infinity. This is one of many ways infinity will receive special treatment. Watch this video for more information.

Order of Operations

Order of operations is a key to understanding how to evaluate an equation. This ranks the operations, addition, multiplication, and so forth, in the order in which they are supposed to be done. Watch the following video to learn more about the order of operations.

If you’re comfortable with order of operations, the interactive component below will create trees showing the correct evaluation order for a variety of equation types.

[WolframCDF source=”http://demonstrations.wolfram.com/OrderOfOperationsTree/OrderOfOperationsTree.cdf” width=”800″ height=”650″]

Scientific Notation

Scientific notation provides a shortcut to writing very large and very small numbers. It is based on the premise that when using numbers in a scientific context, measurements may be very precise (meaning they have a lot of detail, such as 3.141592535) but not necessarily very accurate (meaning, the number may not be as close to the true value as the detail suggests). But in general, you can trust the first several digits, up to a point you decide is important.

So when using scientific notation, there first digit of a number is given, followed by a decimal, and then some remaining digits as appropriate. This is multiplied by 10 to the power necessary to reach the magnitude of the original number. So 3.14 doesn’t change and the year 2011 is $2.011 \times 10^3$. The two example boxes below convert between standard notation and scientific notation. Use both to show yourself how numbers convert between the two forms.

The next interactive will allow you to create your own scientific notation numbers

[WolframCDF source=”http://demonstrations.wolfram.com/ScientificNotation/ScientificNotation.cdf” width=”800″ height=”500″]

Linear Functions

Functions are the idea that we can take a set of things and for each member of that set, there is a member of another set which it corresponds to. We call this a mapping. In these examples, the two sets of things are both the sets of numbers. So a function takes a number, which we will call $x$ and shows how it corresponds to another number, which we will call $y$. We often use the notation $f(x)$ to represent a function. We might also use the notation $y =$ to show the mapping between $x$ and $y$.

Let’s consider this linear function:
$f(x) = 2x + 3$

This function maps the input, any number, to the output which is two times the input plus three. To show the value of this function where, for instance, $x=4$ we use $f(4)$and we know this is $2x+3$ or 11. And there’s a value of the function for any value of $x$:

 $x$ $f(x)$ -2 -1 -1 1 0 3 1 5 2 7

Watch this video for information on recognizing linear functions.

This video includes some more examples of linear functions.

Check out the interactive example below to experiment with a linear function.

[WolframCDF source=”http://demonstrations.wolfram.com/LinearFunctionQuiz/LinearFunctionQuiz.cdf” width=”800″ height=”675″]