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Absolute value, represented by , is the distance from to the origin. Thus, for a real number:
For a complex number:
An arithmetic sequence is a sequence of numbers in which each next term is obtained by adding to the preceding term some fixed difference, .
Example: With initial term and difference , we get the arithmetic sequence:
(Of a function) The argument of a function is the input variable.
Example: For both and are arguments.
(Of a complex number) The argument, of a complex number is the angle between the vector representing the complex number in the complex plane and the positive real axis. This is denoted by . The principle value of the argument is taken to be the unique value of such that .
Example: For the complex number , .
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 , where is a bijective function, since every value in the domain corresponds to exactly one value in the codomain (and vice versa).
The cartesian plane is the 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.
Represented by , the ceiling function (or least integer function) gives the smallest integer greater than or equal to
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.)
A coefficient is a number or constant by which a term involving variablies in an algebraic expression is being multiplied.
Example: In the expression the terms have the coefficients and respectively.
A complex number is a number in the form , where and are real numbers and is the imaginary unit.
Example: is a complex number and so is as it can be written as
The conjugate of a binomial is obtained by multiplying the second term by . The conjugate of a complex number , which equals is sometimes written as .
Example: The conjugate of is . The conjugate of is .
A constant function is one whose value remains the same for all inputs.
Example: is a constant function.
(Of an angle) One degree is 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: has degree 7. has degree .
The discriminant of a polynomial, often denoted , is an expression comprised of the coefficients of the polynomial that gives information about its roots.
Example: The discriminant of the quadratic polynomial is .
Two objects are distinct if they are not exactly the same.
Example: The elements of the set are distinct, but the elements of the set are not.
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 would be all real numbers where . The domain of the function is explicitly given by the set .
An equation asserts the equivalence of two mathematical expression by means of the equal sign: .
Example: is an equation.
An expression is a combination of numbers, operations, and/or variables.
Example: is an expression, as is .
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 we might subtract and square both sides. This would give us . However, is not a valid solution to the original problem since .
Represented by , the floor function (also called the greatest integer function) gives the greatest integer less than or equal to .
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 .
A geometic sequence is a sequence of numbers in which each next term is obtained by multiplying the preceding term by some ratio, .
Example: With initial term and a common ratio of , we get the geometric sequence:
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.
An identity is an equation that is true for every possible value of the variables.
Example: is an identity because it is true for every value of , but is not since it is only true when .
The imaginary unit is represented by the symbol , such that .
An inequality is a statement that relates two expressions that are not, or may not be, equal.
Example: is an inequality that restricts the values of to those which are strictly greater than 5. is an inequality which restricts values to those less than or equal to .
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 .
Example: The following are all integers:
An interval is a connected set of real numbers.
Example: If is in the interval then . If is in the interval then . Note the important difference between "[" and "(".
The leading coefficient of a polynomial is the coefficient of the term with highest degree.
Example: In the polynomial , the term with the highest degree is so the leading coefficient is .
If then is the logarithm of to the base , which is written as .
Example: because .
A monic polynomial is a polynomial with leading coefficient equal to 1.
Example: is a monic polynomial.
A multiset is a set in which one object may appear multiple times.
Example: is a multi with 5 elements.
The natural numbers, denoted by , are the set of numbers such that:
The natural numbers are thus equivalent to the positive integers.
NOTE: Some mathemticians use the definition above but change the first condition to be " 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.
A real number is called negative if .
Example: is a negative number.
The origin is the point in the Cartesian plane, where the -axis and the -axis intersect.
An ordered pair is a set of elements in which the order of the elements is significant. Ordered pairs are often used to represent points in 2-dimensional space.
Example: is an ordered pair, as is . Note that and are distinct because their elements are in a different order.
An ordered triple is a set of elements in which the order of the elements is significant. Ordered triples are often used to represent points in 3-dimensional space.
Example: is an ordered triple, as is . Note that and are distinct because their elements are in a different order.
A polynomial in is an algebraic expression of the form
where the indices are non-negative integers.
Example: is a polynomial in because the indices are non-negative integers. So, is , because the index of is [a non-negative integer]. However is not a polynomial in because the indices of are and respectively, neither of which is a non-negative integer.
A real number is called positive if .
Example: is a positive number.
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.
The range, also called the image, is the set of output values actually taken on by a function.
Example: For the function where , the range would be the set of all non-negative real numbers.
A rational number is any number that can be written as a fraction where and are both integers and . The set of rational numbers is denoted by .
Example: The following are rational numbers: . However, is not a rational number.
Real numbers are those which correspond to the points on a continuous straight line. The real numbers, represented by , are a strict superset of the rational numbers ().
Example: The following are real numbers: . However, complex numbers with a non-zero imaginary part (like ) are not real numbers.
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 . This gives the first few terms .
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 are and .
A sequence is an ordered set of objects (often numbers).
A set is a well-defined group of objects.
Example: The set of all positive integers less than 7 consists of .
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: where is a surjective function, since takes on every value in .
A -tuple is an ordered set with elements.
Example: is a 5-tuple. is a 2-tuple, also called an ordered pair.
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: and are variables in the expression .
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