We have learned to calculate with variables instead of numbers in this chapter. It means that we have to consider things that are not really obvious when you are working with numbers. For example, you can formulate laws for general expressions. Remember the distribution laws and the formula for expanding brackets.

Interesting to note is that the theory of integers and rational numbers arises in a more general form:

• polynomials, sums of products of variables and constants, correspond with integers
• fractions with polynomials as numerators and denominators, correspond with rational numbers.

Many notions from the chapter Numbers, such as denominators, common denominators, and the addition of fractions, have been discussed again. A new subject was partial fraction decompositions; this could have been discussed with integers: by means of #\frac{1}{6}=\frac{1}{2}-\frac{1}{3}# we write the fraction with denominator #6# as a sum of fractions where real divisors of #6# occur as fractions. We did not show a full method to carry through with the partial fraction decomposition. Before we are able to do that, we have to know more about solving linear equations. We will deal with this in the chapters Linear equations with one unknown and Systems of linear equations. Partial fraction decomposition is useful for the integration of functions, a subject which will be covered in the chapter Integration.

Formulas which are usefull for the expanding of brackets, give reason to numbers which show up as coefficients in the formulas and they represent certain numbers. We got acquainted with

• the binomial coefficient #\binom{n}{k}#, which expresses in how many different ways you can choose #k# balls from a black box with #n# balls without repetition and without regarding the order you have chosen the #k# balls.
• the factorial of #n#, which expresses in how many ways you can arrange the numbers #1,2,\ldots,n# in a row.

These numbers occur in solutions to the following problems:

As the examples above show, variables help to formulate expressions which can solve counting problems. However, variables can also be used to mathematically formulate computational problems. A standard example of this is the following:

Pete is 3 years older than Mike. Together they are 101 years old, how old is Pete?

Denoting the age of Pete by #x#, we can write the data as

• the age of Mike is #x+3#, and
• #x+\left(x+3\right) =101#.

The second line of this mathematical formulation is the key to the solution. It is called a linear equation with unknown #x#. In the chapter Linear equations with one unknown, we will discuss a solution for this problem. It is a recommended chapter for a later study.