Appendix A: Basic Mathematics and Related Skills

Negative numbers are preceded by a minus sign. Although the physical quantities used in biomechanics do not have values that are less than zero in magnitude, the minus sign is often used to indicate the direction opposite the direction regarded as positive. Therefore, it is important to recall the following rules regarding arithmetic operations involving negative numbers:

1. 1. Addition of a negative number yields the same results as subtraction of a positive number of the same magnitude:

1. 2. Subtraction of a negative number yields the same result as addition of a positive number of the same magnitude:

1. 3. Multiplication or division of a number by a number of the opposite sign yields a negative result:

1. 4. Multiplication or division of a number by a number of the same sign (positive or negative) yields a positive result:

Exponents are superscript numbers that immediately follow a base number, indicating how many times that number is to be self-multiplied to yield the result:

Taking the square root of a number is the inverse operation of squaring a number (multiplying a number by itself). The square root of a number is the number that yields the original number when multiplied by itself. The square root of 25 is 5, and the square root of 9 is 3. Using mathematics notation, these relationships are expressed as the following:

Because −5 multiplied by itself also equals 25, −5 is also a square root of 25. The following notation is sometimes used to indicate that square roots may be either positive or negative:

When a computation involves more than a single operation, a set of rules must be used to arrive at the correct result. These rules may be summarized as follows:

1. 1. Addition and subtraction are of equal precedence; these operations are carried out from left to right as they occur in an equation:

1. 2. Multiplication and division are of equal precedence; these operations are carried out from left to right as they occur in an equation:

1. 3. Multiplication and division take precedence over addition and subtraction. In computations involving some combination of operations not of the same level of precedence, multiplication and division are carried out before addition and subtraction are carried out:

1. 4. When parentheses (), brackets [ ], or braces { } are used, the operations enclosed are performed first, before the other rules of precedence are applied:

Simple computations in biomechanics problems are often performed quickly and easily with a hand-held calculator. However, the correct result can be obtained on a calculator only when the computation is set up properly and the rules for ordering of operations are followed. Most calculators come with an instruction manual that contains sample calculations. It is worthwhile to completely familiarize yourself with your calculator’s capabilities, particularly use of the memory, before using it to solve problems.

A percentage is a part of 100. Thus, 37% represents 37 parts of 100. To find 37% of 80, multiply the number 80 by 0.37:

The number 29.6 is 37% of 80. If you want to determine the percentage of the number 55 that equals 42, multiply the fraction by 100%:

The number 42 is 76.4% of 55.

The solution of many problems involves setting up an equation containing one or more unknown quantities represented as variables such as x. An equation is a statement of equality implying that the quantities expressed on the left side of the equals sign are equal to the quantities expressed on the right side of the equals sign. Solving a problem typically requires calculation of the unknown quantity or quantities contained in the equation.

The general procedure for calculating the value of a variable in an equation is to isolate the variable on one side of the equals sign and then to carry out the operations among the numbers expressed on the other side of the equals sign. The process of isolating a variable usually involves performing a series of operations on both sides of the equals sign. As long as the same operation is carried out on both sides of the equals sign, equality is preserved and the equation remains valid:

Subtract 7 from both sides of the equation:

Add 3 to both sides of the equation:

Divide both sides of the equation by 3:

Multiply both sides of the equation by 4:

Subtract 5 from both sides of the equation:

Multiply both sides of the equation by 3:

Add 7 to both sides of the equation

Multiply both sides of the equation by 4:

Take the square root of both sides of the equation:

The following procedure is used for measuring an angle with a protractor:

1. 1. Place the center of the protractor on the vertex of the angle.

2. 2. Align the zero line on the protractor with one of the sides of the angle.

3. 3. The size of the angle is indicated on the protractor scale where the other side of the angle intersects the scale. (Be sure to read from the correct scale on the protractor. Is the angle greater or less than 90˚?)

If you are unfamiliar with the use of a protractor, check yourself by verifying the sizes of the following three angles:

(Answer: A = 45˚, B = 60˚, C = 123˚)