# Algebra Topics

The operation of finding the sum of two or more quantities.
The additive identity is the number zero, because zero will not change a number when added to it: a + 0 = a for all a.
The addition of number and it′s additive inverse is zero. The additive inverse of a number a is -a, also known as the opposite of a, such that a + (-a) = 0. For example, 1 + (-1) = 0.
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Algebraic
Defined by the basic algebra operations of addition, subtraction, multiplication (including powers), and division.
Algebraic Equation
An equation of the form f(x)=0 where f is a polynomial.
Algebraic Number
A number that is the root of an algebraic polynomial.
AND
In Boolean algebra, the operation of intersection.
Antilogarithm
x=by is called the antilogarithm of y to the base b.
Basis Set
A set of mathematical functions that are combined to approximate the wavefunctions for electrons in atoms and molecules.
Binary Operation
An operation that involves two operands.
Binomial
An expression that is the sum of two terms.
Binomial Theorem
This gives the form of the expansion of any positive integral power of a binomial (x+a)n as a polynomial.
A polynomial equation of the 4th degree.
Briggsian Logarithm
Another name for the Common Logarithm.
Cauchy-Schwarz Inequality
The dot product of two vectors cannot be greater in magnitude than the product of the magnitudes of the vectors.
Characteristic Polynomial of a Matrix
The characteristic polynomial of a n by n matrix A is the polynomial in t given by the formula det(A - tI).
Column Space of a Matrix
The subspace spanned by the columns of the matrix considered as a set of vectors.
Common Denominator
A multiple shared by the denominators of two or more fractions.
Common Fraction
A fraction whose numerator is an integer of smaller value than its denominator.
Common Logarithm
Logarithm in which the base is 10.
Complex Numbers, Logarithms of
A complex number expressed in logarithmic form.
Connected Set
A set that cannot be split into a union of two sets each of which is both open and closed.
Continuum
Any set that may be brought into 1-1 correspondence with the set of real numbers. Examples: a finite line segment, a square, a circle, a disk.
Cubic Equation
An equation of the third order.
Degree
A unit describing a plane angle, 1 degree = 1/90 right angle.
Denominator
The quantity or value on the bottom of a fraction.
Descartes Rule of Signs
The number of positive real zeros of a polynomial is either equal to the number of variations in sign of the coefficients or else less than that number by a positive even integer.
Difference
The difference between two numbers is what you get when you subtract one from the other.
Division
The binary operation of finding the quotient of two quantities.
Dot Product
The dot product of two vectors is obtained by adding the products of the respective components of the vectors.
Egyptian Fraction
A number of the form 1/x where x is an integer is called an Egyptian fraction.
Eigenvector of a Matrix
An eigenvector of a square matrix A is a nonzero vector x such that Ax = cx holds for some scalar c.
Elementary Function
One of the functions: rational functions, trigonometric functions, exponential functions, and logarithmic functions.
Empty Set
The set with no elements in it.
Enumerable Set
A countable set.
Envelope of a Function
A curve that is tangent to the peaks of the function.
Equation
A statement that two expressions are equal to each other.
Exponential Constant
The base of natural (Napierian) logarithms e = 2.718281...
Exponential Smoothing
A statistical technique commonly used to forecast time series data or to smooth the values on a control chart.
Factor Theorem
The factor theorem states that a polynomial f(x) has a factor (x-k) if and only if f(k) = 0.
Finite Cardinals
Just regular non-negative integers.
Floor Value
For a real number r, its floor value [r] is defined as the largest integer no greater than r. Thus [5]=[5.1]=5 and [-5]=-5 while [-5.1]=-6.
Fraction
A ratio of two integers, or any number that can be expressed as such a ratio.
Fractional Part
For a real number r, its fractional part is defined as {r}=r-[r], where [r] is the floor value of r.
Group
A vertical column in the periodic table.
Growth
Functions may grow monotonously or in jumps. Complexity of a system may grow exponentially with the system size.
Heredity
A property of a space is hereditary if every of its subspaces possesses this property.
Improper Fractions
A fraction whose numerator is of greater absolute value than it's denominator.
Injection
A one to one mapping.
Intersection of Sets
The intersection of two or more sets is the set of elements that all the sets have in common.
Latent Vector
Another name for Eigenvector.
Least Common Multiple
The least common multiple of a set of integers is the smallest integer that is an exact multiple of every number in the set.
Linear Equations
A first order differential equation is linear if it has the form: y′=A(x)B(y)
Linear Function
A function of the form y=ax+b.
Logarithm
The power to which a base must be raised to yield a given number.
Logarithm of Complex Numbers
A complex number expressed in logarithmic form.
Lowest Common Denominator
The smallest number that is exactly divisible by each denominator of a set of fractions.
Magnitude
The size of a vector quantity.
Multinomial
An algebraic expression consisting of 2 or more terms.
Multiple
The integer b is a multiple of the integer a if there is an integer d such that b=da.
Multiples
The product of multiplying a number by a whole number. The multiples of 3 are 6, 9, 12, 15,…
Multiplicand
In the equation x = mn, m and n are the multiplicands since they are the two objects being multiplied together.
Multiplication
In arithmetic, multiplication of one number, a, by another, b, consists of adding a to itself b times.
Multiplication Factor
The number of times something is multiplied.
Multiplication Rule
The probability that events A and B both occur, is equal to the conditional probability that A occurs given that B occurs, times the unconditional probability that B occurs.
Multiplicative Identity
The number 1 is the multiplicative identity.
Multiplicative Inverse
The number that when multiplied by the original number will result in a product of one.
Napierian Logarithm
Another name for the Natural or Common Logarithm.
Natural Base of Logarithms
e = 2.71828…
Natural Logarithm
Logarithm in which the base is e.
Null Set
The null set is a set that contains no objects. Also known as the empty set.
Null Space of a Matrix
The null space of a m by n matrix A is the set of all vectors x in Rn such that Ax = 0.
Nullity of a Matrix
This is the dimension of its null space.
onto
A function f is said to map A onto B if for every b in B, there is some a in A such f(a)=b.
Orthogonal Complement of a Subpace
The orthogonal complement of a subspace S of Rn is the set of all vectors v in Rn such that v is orthogonal to every vector in S.
Proper Fractions
A fraction whose numerator is of lower absolute value than it's denominator.
An equation of the second order.
An algebraic expression consisting of 4 terms.
Rational Function
Any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
Ring
An additive commutative group in which a second operation (normally considered as multiplication) is also defined.
Round Off
To delete less significant digits from a number and possible apply some rule of correction to the part retained.
Row Space of a Matrix
The subspace spanned by the rows of the matrix considered as a set of vectors.
Scalar
Any quantity that has only magnitude as opposed to both magnitude and direction.
Set
A collection of objects called elements.
Similar Matrices
Matrices A and B are similar if there is a square invertible matrix S such that S-1AS = B.
Simple Fraction
A fraction whose numerator is an integer of smaller value than its denominator.
Square Number
A number of the form n2.
Subgroup
A subset H of a group G is a subgroup (of G) provided it′s a group with respect to the group operation of G.
Subset
A subset of a given set is a collection of things that belong to the original set.
Subspace
A subset W of Rn is a subspace of Rn if….
Subtraction
The binary operation of finding the difference between two quantities.
Transfinite Cardinals
Defined as collections (equivalency classes) of sets that could be put into a 1-1 correspondence with each other.
Trinomial
An algebraic expression consisting of 3 terms.
Union of Sets
The union of two or more sets is the set of all the objects contained by at least one of the sets.
Unit Fraction
A fraction whose numerator is 1.
Unit Vector
A vector with a length of 1.
Vector
A quantity with a magnitude and a direction.
Vector Space
The three dimensional area where vectors can be plotted.
Vulgar Fraction
A rational number expressed as a ratio rather than as a decimal fraction.
Whole Numbers
The whole numbers is the set of natural number plus zero, that is 0, 1, 2, 3, 4, 5, . . .
Zero
In 876BC a symbol to represent zero was first used in India.
Zero Divisors
Nonzero elements of a ring whose product is 0.