Mathematics Terminology Study Guide for McGraw-Hill's Firefighter Exams (page 2)
Many firefighter exams include mathematical problems pertaining to the job of firefighting. Knowledge of basic arithmetic, algebra, and geometry is necessary in the fire service. Measuring gas/oil mixtures that fuel portable power saws, mixing cleaning fluids with water during maintenance chores, determining the placement of a ladder for proper climbing angle, and monitoring gallons per minute (GPM) flowing through hose lines are just a few examples of how firefighters use numbers. The mathematics selected for this review includes the most common areas included in previous firefighter examinations.
Basic Terminology, Symbols, and Order of Operations
Terminology (The Language of Mathematics)
There are many terms used in mathematics that you should understand. A list of some of the more common terms and their definitions follows:
Real Numbers: both rational and irrational numbers.
Whole Numbers: the counting numbers that include zero.
Example: 0, 1, 2, 3, ...
Natural (Positive) Numbers: all whole (counting) numbers greater than zero.
Example: 1, 2, 3, 4, …
Negative Numbers: all numbers less than zero.
Example: -1,-2,-3, …
Integers: all positive and negative whole numbers, including zero, but not including fractions and decimals.
Example: -2,-1, 0, 1, 2, …
Fraction: part of a whole number.
Mixed Number: a whole number and a fraction.
Rational Numbers: all numbers that can be expressed as the ratio of two integers (fractions and integers).
Example:x/y where xand y are integers
Irrational Numbers: all numbers that cannot be expressed as the ratio of two integers.
Example: π (3.14) and many square roots, etc.
Term: a single number or the product of one or more numerical (number) coefficient and/or literal (letter) coefficient factors. Like terms have the same literal factor. Unlike terms do not have a common literal factor.
Example: (like terms): 2a and 3a.
Example: (unlike terms): 6, 3x, and yz
Factor: an individual number (numerical) or letter (literal) in a term.
Example: 2, A, and b are factors of the term 2ab
Algebraic Expression: two or more terms connected by plus or minus signs.
Example: 4k + 7
Algebraic Equation: a statement representing two things that are equal to one another.
Example: 12x – 8x = 4x
Mathematical Symbols and Corresponding Common Phrases
Order of Operations (Evaluating Expressions)
Many numerical expressions include two or more operations—for example, exponents, division, and addition. These operations must be performed in correct sequential order. The acronym PEMDAS helps in remembering what the correct sequence is.
PEMDAS—Please Excuse (My Dear) (Aunt Sally)—stands for P arenthesis, E xponents, M ultiplication/D ivision, then A ddition/S ubtraction.
When there are two or more Parentheses, grouping symbols, perform the innermost grouping symbol first. Exponents should be worked on next. Multiplication and Division, as well as Addition and Subtraction are grouped to denote that when these operations are next to each other, just perform the math from left to right.
PEMDAS is used when evaluating formulas, solving equations, solving algebraic expressions, and working with monomials and polynomials.
An ordered list of terms in which the difference between consecutive terms is constant is called an arithmetic sequence. If you subtract any two consecutive terms of the sequence you will obtain the same difference, known as the constant interval between the terms.
A statement that one expression is greater than or less than another expression is called an inequality.
Several symbols are used in statements and word problems involving inequalities. A list of these symbols and their meaning follows:
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