ALGEBRAIC OPERATIONS
What are the commontypes of mathematical operations?
Operations are ways to obtain a single entity from one or more entities by manipulat-
ing numbers (and letters) in certain ways. They include the elementary operations
most of us are familiar with: addition, subtraction, multiplication, division, cubing,
squaring, and integer root extraction. There are numerous types of other operations,
too, including binary operations,in which two quantities or expressions xand yinter-
act in set theory (for more about set theory, see “Foundations of Mathematics”).
What is the algebraic concept of inverse?
“Inverses” in algebra are operations (or numbers) that “undo” each other. For exam-
ple, if one multiplies 4 by its inverse, or 1/4, the solution is 1; thus, the rule for multi-
plicative inverse for x(with x 0) is 1/x, as in x(1/x) 1. In addition, if you add 4
to 4, you get zero (0); thus, the additive inverse of xis x, as in x(x) 0.
What are identityand conditional equations?
Identity and conditional equations are ways in which numbers associate with each
other. When an equation is true for every value of the variable, then the equation is
called an identity equation. It is often denoted as Ior E(the Eis from the German
Einheit,or “unity”). For example, 3x 3 xis an identity equation, because xwill
always be the same number. Zero is the identity element for addition, because any
number added to 0 does not change the value of any of the other numbers in the oper-
ation (or x 0 x). The number 1 is the identity element of multiplication, as any
number in an operation multiplied by 1 does not change the value of that number.
Multiplication identity is often written as x 1 x. 141
ALGEBRA
What is a system of equations?
A
system of equations is any set of simultaneous equations—two or more—
that are intertwined and have to be determined together. They are usually a
finite set of equations with the same unknowns, all of which have common solu-
tions. A set of linear equations is said to be a linear system,while a set of homo-
geneous linear equations is called a homogeneous linear system. The number of
equations is finite, meaning the solutions don’t go on forever like some with bil-
lions of answers. But a word of caution: Some problems use systems with hun-
dreds of equations and just about the same number of variables.