Everyday Usage and Use in Mathematics

In mathematics, a variable is a symbol, usually a letter, that represents a number that can vary or change. Variables are fundamental to algebra and are used to express general mathematical relationships and functions.

In everyday language, the word "variable" means literally 'able to be varied'. 'Vary' is a synonym with 'change', but 'change' seems to imply more human action while 'vary' seems less controlled; this isn't definitive, but it is the connotation carried for English speakers.

Independent and Dependent Variables

When dealing with functions and equations, the variables you will encounter are largely split into two kinds: independent and dependent variables.

Dependent Variables

The word 'depend' is used to indicate reliance or need. For example, we might say "You can depend on your friend for help," meaning that you can rely on your friend when you need assistance. Or you might answer any question with "it depends," meaning the answer to that question is different based on another condition.

The value of a dependent variable depends on the inputs from independent variables. The output of a function is a dependent variable.

Independent Variable

'Independent' describes something that stands alone or is not influenced by other things. For example, a country may 'declare independence,' meaning that it decides to rule itself without being controlled by another country. The song "Independent Women" by Destiny's Child describes women who make their own decisions and provide for themselves without relying on anyone else.

The independent variable is the variable that represents the input or cause, and it is the variable that can be freely changed. In a function, the independent variable is the one you choose values for, and it determines the value of the dependent variable.

Summary

A variable in mathematics is a symbol, usually a letter, that represents a number that can change. Variables are crucial in algebra for expressing mathematical relationships. They are often categorized into independent and dependent variables. The independent variable is the one you control or choose, while the dependent variable changes based on the independent variable.

Exercises

1. Identifying Variables: Given the function \( f(x) = 3x - 7 \), identify the independent and dependent variables.
2. Application of Variables: In a science experiment, the amount of sunlight a plant receives (in hours) affects its growth (in centimeters). Identify the independent and dependent variables.
3. Creating a Function: Write a function where the dependent variable is the total cost \( C \) of buying \( n \) apples if each apple costs $0.50. Identify the independent and dependent variables.