Algebra Glossary

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A

<a name="absolute-value"></a>Absolute Value

Absolute value, represented by $\lvert x \rvert$, is the distance from $x$ to the origin. Thus, for a real number:

$|x| = \begin{cases} x &\text{ if }x \ge 0 \\ -x & \text{ if } x < 0. \end{cases}$

For a complex number:

$|a+bi|=\sqrt{a^2+b^2}$

Example: $\lvert -5 \rvert = 5$

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each next term is obtained by adding to the preceding term some fixed difference, $d$.

Example: With initial term $2$ and difference $3$, we get the arithmetic sequence: $2, 5, 8, 11, \dots$

Argument

(Of a function) The argument of a function is the input variable.

Example: For $f(x,y)=x^2 + y^2$ both $x$ and $y$ are arguments.

(Of a complex number) The argument, $\theta,$ of a complex number $z=a+bi$ is the angle between the vector representing the complex number in the complex plane and the positive real axis. This is denoted by $\theta = \arg (z)$. The principle value of the argument is taken to be the unique value of $\theta$ such that $-\pi < \theta \leq \pi$.

Example: For the complex number $z = 4 + 4i$, $\arg (z) = \frac{\pi}{4} = 45^{\circ}$.

B

Bijective Function

A bijective function is a function which is both injective and surjective. That is to say, every element in the codomain of the function is taken as a value of the function exactly once.

Example: The function $f: \mathbb{R} \rightarrow \mathbb{R}$, where $f(x) = x$ is a bijective function, since every value in the domain corresponds to exactly one value in the codomain (and vice versa).

C

Cartesian Plane

The cartesian plane is the $x-y$ coordinate plane, with the two axes perpendicular to each other. Every point on the plane can be represented by an ordered pair of real numbers.

Ceiling

Represented by $\lceil x \rceil$, the ceiling function (or least integer function) gives the smallest integer greater than or equal to $x.$

Example: $\lceil 2.99 \rceil = 3$ and $\lceil -1.1 \rceil = -1$

Codomain

The codomain is the target set of a function. That is to say, it is the set into which the output of the function is constrained to fall. Not every element of the codomain must actually be assumed by the function. (If it is, the function is surjective.)

Coefficient

A coefficient is a number or constant by which a term involving variablies in an algebraic expression is being multiplied.

Example: In the expression $4x + xy -5y$ the terms have the coefficients $4, 1,$ and $-5,$ respectively.

Complex Number

A complex number is a number in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Example: $3+ 4i$ is a complex number and so is $5,$ as it can be written as $5+0i$

Conjugate

The conjugate of a binomial is obtained by multiplying the second term by $-1$. The conjugate of a complex number $a + bi$, which equals $a - bi$ is sometimes written as $\overline{a + bi}$.

Example: The conjugate of $2 + 3i$ is $2 -3i$. The conjugate of $2+x$ is $2 - x$.

Constant Function

A constant function is one whose value remains the same for all inputs.

Example: $f(x) = 7$ is a constant function.

D

Degree

(Of an angle) One degree is $\frac{1}{360}$ of one full revolution.

(Of a polynomial) The degree of a polynomial is the power of the term with the largest exponent. For polynomials with multiple variables, the degree is found by identifying the term with the highest sum of exponents.

Example: $f(x) = 2 + 5x + 15 x^7$ has degree 7. $f(x,y) = xy^2 -2 x^3y^2 + x^8y$ has degree $8+1 =9$.

Discriminant

The discriminant of a polynomial, often denoted $\Delta$, is an expression comprised of the coefficients of the polynomial that gives information about its roots.

Example: The discriminant of the quadratic polynomial $ax^2 + bx + c =0$ is $\Delta = b^2 - 4ac$.

Distinct

Two objects are distinct if they are not exactly the same.

Example: The elements of the set $\{1,2,3\}$ are distinct, but the elements of the set $\{1,2,1\}$ are not.

Domain

The domain is the set of input values for a function. Unless stated otherwise or prohibited by the problem itself, the domain is typically assumed to be the set of real numbers.

Example: The domain of $f(x) = \frac{1}{x}$ would be all real numbers where $x \neq 0$. The domain of the function $g: X \rightarrow \mathbb{R}, \text{ where }X=\{1,2,5,6\}$ is explicitly given by the set $X$.

E

Equation

An equation asserts the equivalence of two mathematical expression by means of the equal sign: $=$.

Example: $3+3x=6$ is an equation.

Expression

An expression is a combination of numbers, operations, and/or variables.

Example: $1+2$ is an expression, as is $\frac{x^6+3x^3-x^{\sqrt2}}{xyz^3}$.

Extraneous Solution

An extraneous solution is a solution that emerges from the process of solving the problem but does not satisfy the conditions of the original problem.

Example: In solving for $\sqrt{2x+7} + 3 = 0$ we might subtract $3$ and square both sides. This would give us $2x + 7 = 9 \implies x = 1$. However, $x = 1$ is not a valid solution to the original problem since $\sqrt{2(1)+7} + 3 \neq 0$.

F

Floor

Represented by $\lfloor x \rfloor$, the floor function (also called the greatest integer function) gives the greatest integer less than or equal to $x$.

Example: $\lfloor 2.99 \rfloor = 2$ and $\lfloor -1.1 \rfloor = -2$

Function

A function is an operation or set of operations performed on a set (called the domain), resulting in another set (called the range). For every value in the domain, there can only be one corresponding value in the range.

Example: The operation of "add two" applied to the domain of the real numbers would be a valid function. This would normally be written as $f(x) = x + 2$.

G

Geometric Sequence

A geometic sequence is a sequence of numbers in which each next term is obtained by multiplying the preceding term by some ratio, $r$.

Example: With initial term $2$ and a common ratio of $3$, we get the geometric sequence: $2, 6, 18, 54, \dots$

H

I

Injective Function

An injective function is one where every element in the range is matched up to only one element in the domain. Note that not every element in the codomain must be in the range.

Identity

An identity is an equation that is true for every possible value of the variables.

Example: $4x = x + x + x + x$ is an identity because it is true for every value of $x$, but $3x + 3 = 15$ is not since it is only true when $x = 4$.

Imaginary Unit

The imaginary unit is represented by the symbol $i$, such that $i^2 = -1$.

Example: $\sqrt{-25} = \sqrt{25}\times \sqrt{-1} = 5i$

Inequality

An inequality is a statement that relates two expressions that are not, or may not be, equal.

Example: $x > 5$ is an inequality that restricts the values of $x$ to those which are strictly greater than 5. $x \leq 3$ is an inequality which restricts values to those less than or equal to $3$.

Integer

An integer is any number that can be written with no decimal or fractional part: a natural number, a negative natural number, or zero. The set of all integers is represented by $\mathbb{Z}$.

Example: The following are all integers: $\dots, -3, -2, -1, 0, 1, 2, 3, 4, 5, \dots$

Interval

An interval is a connected set of real numbers.

Example: If $x$ is in the interval $[a,b]$ then $a \leq x \leq b$. If $x$ is in the interval $(a,b]$ then $a < x \leq b$. Note the important difference between "[" and "(".

Image

See RANGE.

J

K

L

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with highest degree.

Example: In the polynomial $3x^2 - 5x^4 +3$, the term with the highest degree is $-5x^4$ so the leading coefficient is $-5$.

Logarithm

If $x = a^y$ then $y$ is the logarithm of $x$ to the base $a$, which is written as $y = \log_a x$.

Example: $\log_2 16 = 4$ because $2^4 = 16$.

M

Monic Polynomial

A monic polynomial is a polynomial with leading coefficient equal to 1.

Example: $f(x) = x^4 + 3x +2$ is a monic polynomial.

Multiset

A multiset is a set in which one object may appear multiple times.

Example: $\{ 1,2,2,2,3\}$ is a multi with 5 elements.

N

Natural Numbers

The natural numbers, denoted by $\mathbb{N}$, are the set of numbers such that:

• $1$ is in the set.
• If $n$ is in the set, $n+1$ is also in the set.

The natural numbers are thus equivalent to the positive integers.

NOTE: Some mathemticians use the definition above but change the first condition to be "$0$ is in the set." Thus the term might denote not "the positive integers" but rather "the non-negative integers". "Natural numbers," therefore, has some ambiguity and should be avoided.

Negative Number

A real number $x$ is called negative if $x < 0$.

Example: $-2$ is a negative number.

O

Origin

The origin is the point $(0,0)$ in the Cartesian plane, where the $x$-axis and the $y$-axis intersect.

Ordered Pair

An ordered pair is a set of $2$ elements in which the order of the elements is significant. Ordered pairs are often used to represent points in 2-dimensional space.

Example: $2, 3$ is an ordered pair, as is $(1,1)$. Note that $(2,3)$ and $(3,2)$ are distinct because their elements are in a different order.

Ordered Triple

An ordered triple is a set of $3$ elements in which the order of the elements is significant. Ordered triples are often used to represent points in 3-dimensional space.

Example: $(2, 3, 1)$ is an ordered triple, as is $(1,1,0)$. Note that $(2,3,1)$ and $(3,2,1)$ are distinct because their elements are in a different order.

P

Polynomial

A polynomial in $x$ is an algebraic expression of the form

$a_nx^n+a_{n-1}x^{n -1}+\cdots a_3x^3 + a_2x^2 + a_1x + a_0,$

where the indices are non-negative integers.

Example: $P(x)=x^2+x+1$ is a polynomial in $x$ because the indices are non-negative integers. So, is $P(x)=5$, because the index of $x$ is $0$ [a non-negative integer]. However $P(x)=\sqrt{x}+\frac{5}{x}$ is not a polynomial in $x$ because the indices of $x$ are $\frac{1}{2}$ and$-1$ respectively, neither of which is a non-negative integer.

Positive Number

A real number $x$ is called positive if $x > 0$.

Example: $2$ is a positive number.

Q

Quadrants

The quadrants are the four parts of the Cartesian plane, not including the axes; they are denoted I, II, III, and IV, beginning in the upper right and proceeding counter-clockwise.

R

Range

The range, also called the image, is the set of output values actually taken on by a function.

Example: For the function $f: \mathbb{R} \rightarrow \mathbb{R}$ where $y= f(x) = x^2$, the range would be the set of all non-negative real numbers.

Rational Number

A rational number is any number that can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are both integers and $q \neq 0$. The set of rational numbers is denoted by $\mathbb{Q}$.

Example: The following are rational numbers: $2, 0, -3, \frac{2}{3}$. However, $\sqrt{2}$ is not a rational number.

Real Number

Real numbers are those which correspond to the points on a continuous straight line. The real numbers, represented by $\mathbb {R}$, are a strict superset of the rational numbers ($\mathbb{Q}$).

Example: The following are real numbers: $2, -3, \frac{2}{3}, 0, \sqrt{3}, \pi$. However, complex numbers with a non-zero imaginary part (like $2 + 3i$) are not real numbers.

Recurrence Relation

A process or sequence in which the next step or term is defined by one or more previous terms.

Example: The Fibonacci sequence is defined by $F_n=F_{n-1}+F_{n-2}, F_0=0, F_1=1$. This gives the first few terms $0,1,1,2,3,5,8,13,...$.

Root

The roots (also called the zeroes) of a polynomial are the values that make the polynomial equal to zero.

Example: The roots of the polynomial $x^2-5x+6 = (x-3)(x-2)$ are $x = 2$ and $x =3$.

S

Sequence

A sequence is an ordered set of objects (often numbers).

Set

A set is a well-defined group of objects.

Example: The set of all positive integers less than 7 consists of $\{ 1,2,3,4,5,6 \}$.

Surjective Function

A surjective function is a function where every element in the codomain has at least one corresponding element in the domain. In a surjective function, the codomain and the range are equivalent.

Example: $f: \mathbb{R} \rightarrow \mathbb{R}$ where $f(x) = x^3 - 3x$ is a surjective function, since $f(x)$ takes on every value in $\mathbb{R}$.

T

Tuple

A $k$-tuple is an ordered set with $k$ elements.

Example: $(1,1,2,4,1)$ is a 5-tuple. $(5,4)$ is a 2-tuple, also called an ordered pair.

U

V

Variable

A variable is a symbol, often a letter, which is used to represent a value which may change within the context of the given problem.

Example: $x$ and $y$ are variables in the expression $y = x^2 +4$.

W

X

Y

Z

Special thanks to these Brilliant members for their help:

Christopher B, Ton D., Mursalin H., Bob K., Mindren L., Shivanshu M., Ahaan R., Michael T., Jonathan W., and Justin W.

View the discussion where they contributed.