# Algebra Topics

**Addition**- The operation of finding the sum of two or more quantities.
**Additive Identity**- The additive identity is the number zero, because zero will not change a number when added to it: a + 0 = a for all a.
**Additive Inverse**- 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.
**Algebra Books**- Lists all Algebra Books in the Encyclopaedia
**Algebra Calculations**- Lists all Algebra Calculations in the Encyclopaedia
**Algebra Conversions**- Lists all Algebra Conversions in the Encyclopaedia
**Algebra Weblinks**- Lists all Algebra Weblinks in the Encyclopaedia
**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=b
^{y}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. **Biquadratic Equation**- 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 R
^{n}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 R
^{n}is the set of all vectors v in R^{n}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.
**Quadratic Equation**- An equation of the second order.
**Quadrinomial**- 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 n
^{2}. **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 R
^{n}is a subspace of R^{n}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.

**See also: **Algebraic.

**Subjects: ** Mathematics