8 Using and Abusing Variables

One fundamental difference between mathematical communication and most non-mathematical communication, is the pervasive use of variables: letters, subscripted letter, or other symbols that represent mathematical objects that are unspecified or not fully specified. Variables provide the skilled mathematical communicator with an enormously powerful tool to communicate complex ideas effectively. But when used improperly, variables can render mathematical communication incomprehensible or nonsensical. The use of variables is governed by rules that are intended to ensure that the resulting communication is correct, meaningful and understandable. Failure to follow these rules can cause great confusion, and result in serious errors such as claiming that you’ve proved something that’s false. As with other proof-writing guidelines presented in this course, the guidelines for using variables are stricter than those used by mathematicians and in most other textbooks. Following the guidelines will protect you against incorrect arguments, and should be followed until you have enough experience to reliably determine when the rules can be safely relaxed.