Variables

Mathematics often uses letters in place of numbers to illustrate a general principle. In a mathematical equation, the letters are referred to as variables because they can represent any number or numbers that can change depending on the situation. There are two types of variables; independent and dependent. An independent variable can change its value, usually without any restrictions. A dependent variable's value usually depends on the value of the independent variables in an equation. If a number alone is present in an equation, it represents a constant or a value that doesn't change.

Letters represent numbers sort of like letters can represent other letters in a code. If the letter A represents F in a code, the letter A represents F everywhere it appears. Similarly, if the letter a represents the number 5 in an equation, it represents 5 everywhere in the equation. The letter b would indicate another number besides 5.

Variables always have a combination of letters and numbers. Numbers in front of variables like 3a means there are three a's in the equation. If no number is in front of a variable it is understood that it is preceded by 1. Exponents are also always present with variables. If they're not explicitly shown they're undertood to be 1.

Variables can behave like numbers. Any operation you can perform on a number can be theoretically performed on a variable.

Multiplying a variable by itself changes the variables's dimension. If the variable a represents a line, then a² represents a square and a³ represents a cube and so forth. You can't add a, a² and a³ because they're not the same shape. You can only add lines to other lines, squares to other squares and cubes to other cubes. You can't add a² and b² because a² is a square and b² is a circle. To add variables together, they have to be the same letter and the same exponent. If the letters are the same and the exponents are the same, you add the numbers that come in front.

3a + 2a = 5a

When you multiply variables of the same kind together, you add the exponents. When dividing variables of the same kind, you subtract the exponent of the second variable from the first.

a²b × ab = a³b²

In a rational expression, a fraction involving variables and numbers, to add or subtract, the numbers and letters and exponents on the letters must be the same. If you have a² in one denominator and a³ in the other denominator, you multiply the first rational number by a/a. Remember that a fraction where the numerator and the denominator are the same is the same as 1, and multiplying a number by 1 doesn't change the value of the number. In this case, we only change the way the number looks.

1/a²b + 2/a³ =
(a/a × 1/a²b) + (b/b × 2/a³) =
a/a³b + 2b/a³b

Now that the denominators are the same we can add the two rational expressions. But notice that the numerators have different numbers and letters.

a/a³b + 2b/a³b = (a + 2b)/a³b

How would this work if we replaced the letters with numbers? Let's let a=3 and b=5.

1/3²5 + 2/3³ =
(3/3 × 1/3²5) + (5/5 × 2/3³) =
3/3³5 + 10/3³5 =
3/135 + 10/135 = 13/135

We can let a and b represent any two different numbers and we'll get a different result each time. Using the numbers only works for a specific case, but using variables allows us to talk about what happens in a general case. This is what variables are used for. We build mathematical sentences to describe what happens in a general sense using variables and then use specific numbers to explore a specific case. We can use variables to describe a general case of how long it takes to get to a destination at any speed, then put in numbers to see the results at specific speeds. To define how long it takes to get to a destination at any given speed we would write an equation using variables for distance and speed. If we let d represent the distance we want to travel and m represent the miles we travel in an hour, our equation would be d × 1/m.

Let's see what happens if we want to travel 100 miles at 55 mph. We put 100 in place of d and 55 in place of m.
100 × 1/55 = 1.812 hrs

To see what happens if we want to travel the same 100 miles at 65 mph we put 100 in place of d and 65 in place of m.
100 × 1/65 = 1.538 hrs

Using variables, we've set up a general case for how long it takes, then we can look at specific cases by substituting numbers for the variables. This is called modeling and is a very important topic in mathematics. All kinds of things can be modeled from the growth of tiny bacteria to landing a man on the moon. There is a famous equation developed by Isaac Newton to determine the gravity exerted by a body in space. All you have to do is substitute the size of the bodies and the distance between them. The same equation works everywhere in the universe, on Earth, on the Moon even on Jupiter or Mars. We have one equation and we can take it anywhere we want to go and substitute numbers for the variables depending on where we are.