Everyday Usage and Use in Mathematics

Algebra is a kind of mathematics that deals with letters, symbols, and rules for using those symbols. The Oxford English Dictionary defines Algebra[1] as "The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formula[s] and equations." These symbols represent numbers and operations and are used to express general mathematical relationships and solve equations. In everyday use, we might say, "We need to use algebra to solve this problem", "My notebook is full of algebra", or "I left my algebra homework at home."

Variables and Algebraic Symbols

In algebra, letters of the alphabet are used to represent numbers that can vary. A letter that represents a changing number is called a variable. Variables allow us to write general expressions and equations that can be solved for different values. For example, we can use \( x \) to represent an unknown number in an equation.

The letters most commonly used for variables are \( x, y, a, b, \) and \( c \), but most letters can be used. Sometimes Greek letters are used as well, often representing a specific variable within a formula.

Variable: A variable is a letter that represents a number whose value may change.

Constant: A constant is a number whose value always stays the same.

Expressions and Equations

Expression: An expression is a number, a variable, or a combination of numbers and variables using operation symbols. For example:

• \( 3 + 5 \) (the sum of three and five)
• \( n - 1 \) (the difference of n and one)
• \( 6 \cdot 7 \) (the product of six and seven)

Equation: An equation is two expressions connected by an equal sign. For example:

• \( 3 + 5 = 8 \) (The sum of three and five is equal to eight)
• \( n - 1 = 14 \) (n minus one equals fourteen)

Exponents

Suppose we need to multiply 2 nine times. We could write this as \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \). This is tedious, so we use exponents. We write \( 2 \cdot 2 \cdot 2 \) as \( 2^3 \) and \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \) as \( 2^9 \).

Exponential Notation: \( a^n \) means multiply \( a \) by itself \( n \) times.

Combining Like Terms

Term: A term is a constant, or the product of a constant and one or more variables. For example: \( 7, y, 5x^2, 9a, \text{and} b^5 \).

Coefficient: The coefficient of a term is the constant that multiplies the variable in a term. For example, in the term \( 3x \), 3 is the coefficient.

Like Terms: Terms that are either constants or have the same variables raised to the same powers.

Exercises

Example: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

" The value of \( x \) is equal to negative \( b \) plus or minus the square root of \( b \) squared, minus \( 4 \) times \( a \) times \( c \), all divided by \( 2 \) times \( a \) "
or
" \( x \) equals negative \( b \) plus or minus the square root of \( b \) squared, minus \( 4 a c \), all over \( 2 a \) "

1. Translating English to Algebraic Expressions

1. " Three \( x \) plus \( 5 \) squared. "
2. " The sum of \( 17 \) and \( 5 \) over \( 2 y \) cubed "

2. Translating Algebra to English:

   1. \( 5 + 3 \)
   2. \( 2x - 4 \)

References

[1] Oxford English Dictionary, s.v. “algebra, n.”, https://www.oed.com/dictionary/algebra_n