Algebra Help Math Sheet

An Engineers Quick Algebra Reference

$algebra sheet$

Algebra Math Help

Arithmetic OperationsReturn to Top

The basic arithmetic operations are addition, subtraction, multiplication, and division. These operators follow an order of operation.

AdditionReturn to Top

Addition is the operation of combining two numbers. If more than two numbers are added this can be called summing. Addition is denoted by + symbol. The addition of zero to any number results in the same number. Addition of a negative number is equivalent to subtraction of the absolute value of that number.

SubtractionReturn to Top

Subtraction is the inverse of addition. The subtraction operator will reduce the first operand (minuend) by the second operand (subtrahend). Subtraction is denoted by - symbol.

MultiplicationReturn to Top

Multiplication is the product of two numbers and can be considered as a series of repeat addition. Multiplication of a negative number will result in the reciprocal of the number. Multiplication of zero always results in zero. Multiplication of one always results in the same number.

DivisionReturn to Top

Division is the method to determine the quotient of two numbers. Division is the opposite of multiplication. Division is the dividend divided by the divisor.

Arithmetic PropertiesReturn to Top

The main arithmetic properties are Associative, Commutative, and Distributive. These properties are used to manipulate expressions and to create equivalent expressions in a new form.

AssociativeReturn to Top

The Associative property is related to grouping rules. This rule allows the order of addition or multiplication operation on numbers to be changed and result the same value.

$associative property$

CommutativeReturn to Top

The Commutative property is related the order of operations. This rule applies to both addition and subtraction and allows the operands to change order within the same group.

$commutative property$

DistributiveReturn to Top

The law of distribution allows operations in some cases to be broken down into parts. The property is applied when multiplication is applied to a group of division. This law is applied in the case of factoring.

$distributive property$

Arithmetic Operations ExamplesReturn to Top

$algebra operations 1$

$algebra operations 2$

$algebra operations 3$

$algebra operations 4$

$algebra operations 5$

$algebra operations 6$

$algebra operations 7$

$algebra operations 8$

$algebra operations 9$

$algebra operations 10$

Exponent PropertiesReturn to Top

$exponent addition property$

$exponent multiplication property$

$exponent multiply base property$

$negative exponent property$

$negative exponent with division property$

$exponent division propery$

$exponent of zero property$

$exponent fraction property$

$invert exponent property$

$fraction exponent$

Properties of RadicalsReturn to Top

$radical property$

$double radical property$

$multiply radical property$

$divide radical property$

$exponent radical propery odd$

$exponent radical propery even$

Properties of InequalitiesReturn to Top

$inequalities subtraction property$

$inequalities division less than property$

$inequalities division greater than property$

Properties of Absolute ValueReturn to Top

$absolute value definition$

$negative absolute value property$

$absolute value zero property$

$absolute value multiply property$

$absolute value divide property$

$absolute value sum propety$

Complex Numbers

Definition of Complex NumbersReturn to Top

Complex numbers are an extension of the real number system. Complex numbers are defined as a two dimension vector containing a real number and an imaginary number. The imaginary unit is defined as:

$imaginary number definition$

The complex number format where a is a real number and b is an imaginary number is defined as:

$complex number format$

Unlike the real number system where all numbers are represented on a line, complex numbers are represented on a complex plane, one axis represents real numbers and the other axis represents imaginary numbers.

Properites of Complex NumbersReturn to Top

$definition of a complex number$

$format of complex numbers$

$property of the square of a complex number$

$property of a negative complex number$

$complex numbers addition property$

$complex numbers subtraction property$

$complex numbers multiplication property$

$complex numbers conjugate property$

$complex numbers absolute value property$

$complex numbers magnitude property$

$complex numbers absolute value squared property$

Logarithms

Definition of LogarithmsReturn to Top

A logarithm is a function that for a specific number returns the power or exponent required to raise a given base to equal that number. Some advantages for using logarithms are very large and very small numbers can be represented with smaller numbers. Another advantage to logarithms is simple addition and subtraction replace equivalent more complex operations. The definition of a logarithms is:

$log and inverse log definition$

Definition of Natural LogReturn to Top

$natural log definition$

Definition of Common LogReturn to Top

$common log definition$

Logarithm PropertiesReturn to Top

$log of number equal to base property$

$log of one property$

$log of base raised to power property$

$base raised to log property$

$log power to multiplication property$

$log multiplication property$

$log divisin property$

Factoring

PolynomialsReturn to Top

A polynomial is an expression made up of variables, constants and uses the operators addition, subtraction, multiplication, division, and raising to a constant non negative power. Polynomials follow the form:

$polynomial definition$

The polynomial is made up of coefficients multiplied by the variable raised to some integer power. The degree of a polynomial is determined by the largest power the variable is raised.

Quadratic EquationReturn to Top

A quadratic equation is a polynomial of the second order.

$quadratic equation$

The solution of a quadratic equation is the quadratic formula. The quadratic formula is:

$quadratic solution$

Common Factoring ExamplesReturn to Top

$quadratic factoring example 1$

$quadratic factoring example 2$

$quadratic factoring example 3$

$quadratic factoring example 4$

$cubic factoring example 5$

$cubic factoring example 6$

$cubic factoring example 7$

$cubic factoring example 8$

Square RootReturn to Top

The square root is a function where the square root of a number (x) results in a number (r) that when squared is equal to x.

$square root definition$

Also the square root property is:

$square root property$

Absolute ValueReturn to Top

$absolute value properties 1$

$absolute value properties 2$

$absolute value properties 3$

Completing the SquareReturn to Top

Completing the square is a method used to solve quadratic equations. Algebraic properties are used to manipulate the quadratic polynomial to change its form. This method is one way to derive the quadratic formula.

$completing the square$

The steps to complete the square are:

Divide by the coefficient a.
Move the constant to the other side.
Take half of the coefficient b/a, square it and add it to both sides.
Factor the left side of the equation.
Use the square root property.
Solve for x.

Functions and Graphs

Expressions evaluated at incremental points then plotted on a Cartesian coordinate system is a plot or graph.

Constant FunctionReturn to Top

When a function is equal to a constant, for all values of x, f(x) is equal to the constant. The graph of this function is a straight line through the point (0,c).

$constant function$