Calculating with letters: Computing with letters
Basic rules
Apply priority rules when you calculate with numbers.
Priority rules
- Addition and multiplication are applied in the order in which they occur, from left to right.
- Multiplication, and division are applied in the order in which they occur, from left to right.
- Multiplication and division have priority over addition and subtraction.
- Exponentiation has priority over multiplication and division
"Please Excuse My Dear Aunt Sally" (PEMDAS) is a mnemonic in the USA to remember some priority rules for calculation: first Parenthesis, then Exponents, hereafter Multiplication, followed by Division, and finally Addition and Subtraction. In other countries similar mnemonics are used: for instance, BIDMAS used in the UK stands for the order Brackets, Indices, Division, Multiplication, Addition, Subtraction
With brackets you can change the normal order of operation.
Variable When you replace the numbers in an arithmetic expression by letters, than you get an algebraic expression. The letters are then like slots for numbers. But they can be symbols for a variable quantity, for example the letter for temperature or for time. Such a letter is called a variable.
The same priority rules are applied in calculations with letters. After all, when you fill in values for the variables, you must be able to interpret the result of substitution as a calculation with number. You can also calculate with letters and for example call the sum of and , or call the doubling of .
But you must be careful in what you do when you substitute numbers and not simply replace for example in by : substitution of in is not equal to , but results in this example in the number . When substituting negative number, place brackets when needed: substitution of in is not equal to , but equal to .
The following notational agreements are used when calculating with letters.
Notational conventions
- Because the multiplication symbol resembles too much the letter it is often replaced in formulas by a point or it is completely omitted when no confusion can arise.
- In mixed forms of numbers and letters, for example , it is common to first write the coefficient: so and not .
- If the notation leaves room for different interpretations or an expression simply is hard to read, one often uses brackets and points as multiplication sign: is by convention and not equal to or .
The expression is a product and the number is called the coefficient of .
The expression is a sum with the termen and . The number is the coefficient of the term .
For any variable and any positive integer we have:
Herewith we defined for any integer . The number is called the exponent. Instead of we write , and we agree on .
Examples
For any variable and any positive integers and we have:
Herewith we defined for any positive rational . The number is called exponent. We agree on
Examples
Simplification
The expression can be simplified into . The expression becomes less wieldy when you replace it by . Making an algebraic expression more simple and shorter is referred to as simplification.
In simplification of algebraic expression you often use the commutative, associative, and distributive properties to
- do calculations with numbers;
- collect similar terms, i.e. terms with the same letters and exponents;
- alphabetically order letters and placing numbers in front.
Examples
Mathcentre videos
Rules of Arithmetic (29:07)
Substitution in Formulae (20:58)
Mathematical Language (21:24)