Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols to solve equations and to represent mathematical structures and relationships.  It is concerned with the study of equations, functions, and their properties, and how they relate to each other.  In algebra, letters and symbols are used to represent unknowns and variables, and equations are used to represent mathematical relationships between these variables.  Algebraic equations are solved by manipulating the variables and operations on them according to a set of rules and principles.

Algebra has many applications in fields such as physics, engineering, economics, and computer science.  It is a fundamental tool for solving problems in these fields, and is also used extensively in pure mathematics research.

• See Articles  -  List of Tags, List of Categories, List of Articles, List of Glossaries, Nomenclature and Symbols

Algebra Types

• Abstract Algebra, Associative Algebra, Category Theory, Differential Algebra, Elementary Algebra, Group Theory, Homological Algebra, Field Theory, Lattice Theory, Lie Algebra, Linear Algebra, Multilinear Algebra, Non-associative Algebra, Ring Theory, Universal Algebra

Algebra Glossary

A

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z

• Abscissa  -  The first coordinate in an ordered pair.  For the point  \(\large{ 3, 7 }\)  the abscissa is  \(\large{ 3 }\) .
• Absolute Value  -  Makes a negative number positive  \(\large{ \left\vert -x \right\vert = x }\)  and positive numbers and  \(\large{ 0 }\)  are not changed.
• Addend  -  Any one of a set of terms  \(\large{ 3 + 7 = 10 }\)  to be added.  \(\large{ 3 }\)  and  \(\large{ 7 }\)  are each addends,  \(\large{ 10 }\) is the sum.
• Algebraic Expression  -  Includes variables, if not, then it is called an arithmetic expression.  Equation is \(\large{13a^2 + 7x = 18 }\)  The variables are \(\large{ a }\) and \(\large{x }\) .
• Algebraic Properties  -

• • A postulate is a statement that is assumed true without proof.
  • A theorem is a true statement that can be proven.

• Associative Law of Addition  -  For any three numbers a, b, and c, it is always true that  \(\large{ (a+b)+c=a+(b+c) }\).
• Associative Law of Multiplication  -  For any three numbers a, b, and c, it is always true that  \(\large{ (a(b))(c)=a(b(c)) }\).
• Associative Property  -  How you group the numbers does not matter.  \(\large{ \left(a+b\right)+c = a+\left(b+c\right) }\)  or  \(\large{ \left(a\;b\right)\;c = a\; \left(b\;c\right) }\)
• Axes  -  A horizintal number line, x-axis and a vertical number line, y-axis.  Both used on a coordinate system or graph.
• Axiom  -  A statement accepted as true without proof.

B

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Base  -  The term  \(\large{13a^2 }\)  has a base  \(\large{ a }\) .
• Binary Number  -  Use only the digits \(\large{ 0 }\) and \(\large{ 1 }\) .
• Binomial  -  A polynomial with only two term  \(\large{ 13a^2+7x }\) .

C

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Cartesian Coordinates (Rectangular Coordinates)  -  \(\large{ x, y }\)  or  \(\large{ x, y, z }\)
• Coefficient  -  A number multiplied by a variable.  An equation  \(\large{13a^2+7x-21=19 }\) , the coefficients are  \(\large{13, 7 }\) .
• Combination  -  A set of objects in which the order is not important.  \(\large{ \left(7, 21, 19\right) }\)  or  \(\large{ \left(19, 7, 21\right) }\)
• Common Demoninator  -  Two or more fractions  \(\large{ \frac{3}{8} + \frac{7}{8}}\)  that have the same denominator  \(\large{ 8 }\) .
• Common Difference  -  \(\large{ 3 }\)  is the difference between each number  \(\large{ 3, 6, 9, 12, ... }\)  in a sequence  \(\large{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... }\) .
• Common Factor  -  The factors of two or more numbers that have some factors that are the same (common) in each.
• Common Fraction  -  A fraction where both numbers  \(\large{ \frac{3}{4}, \frac{7}{8} }\)  top and bottom are integers.
• Common Multiple  -  Two or more numbers that have the same multiple.
• Common Ratio  -  A number multipling the previous term in a geometric sequence.  Series  \(\large{ 3, 6, 12, 24, ... }\)  with a common rario of 2.
• Commutative  -  When the order of the numbers do not matter.  Works for addition and multiplication but not for subtraction or division.    \(\large{ 3 + 7 = 7 + 3 }\)  or  \(\large{ 3\; x\; 7 = 7\; x\; 3 }\)
• Commutative Property  -  The moving aroung of the numbers using  \(\large{ + }\)  of  \(\large{ \times }\)  does not matter.  \(\large{ a + b = b + a }\)  or  \(\large{ a \; b = b \; a }\)
• Comparison  -  Compasring two numbers to see which is the largest.
• Complex Fraction (Compound Fraction)  -  A fraction where the denominator, numerator or both contain a fraction.  \(\large{ \frac{ 5 }{ \frac{7}{8} } }\) ,  \(\large{ \frac{ \frac{3}{8} }{ 9 } }\) ,  \(\large{ \frac{ \frac{3}{8} }{ \frac{7}{8} } }\)
• Complex Number  -  A combination of a real  \(\large{3, \frac{3}{4}, 13.45, -3.56, ... }\)  number and imaginary  \(\large{\sqrt{-1} = i }\)  number for a result of  \(\large{x + y\;i }\) .   \(\large{ x }\)  is the real part and  \(\large{ y }\)  is the imaginary part.
• Composite Number  -  A positive integer number  \(\large{ 4, 6, 8, 9,... }\)  that has factors other than  \(\large{ 1 }\)  and the number itself.
• Compute  -  To compute  \(\large{ 3-2 }\)  is to figuring out the answer  \(\large{ 1 }\) .
• Commutative Law of Addition  -  For any two numbers a and b.  \(\large{a+b = b+a}\)
• Commutative Law of Multiplication  -  For two numbers a and b.  \(\large{ a(b) = b(a) }\)
• Conjugate  -  Is when you change the sign.  from  \(\large{ a+b }\)  to  \(\large{ a-b }\),  from  \(\large{ 3a-4b }\)  to  \(\large{ 3a+4b }\)  \(\large{ ,... }\)
• Consecutive Number  -  Numbers that follow each other in order, from smallest to largest.  \(\large{ 15, 20, 25, 30, 35, ... }\)
• Constant  -  The term expressed with no variables.  An equation  \(\large{13a^2+7x-21=19 }\) , the constants are  \(\large{21, 19 }\) .
• Conversion  -  The act of changing a unit to a different unit of measure.
• Counting Number  -  Any number used to count things  \(\large{ 1, 2, 3, 4, 5, 6,... }\)  excluding  \(\large{ 0 }\) , negative numbers, fractions or decimals.
• Cube Number  -  \(\large{ 5 \times 5 \times 5 = 5^3 = 125 }\) ,  \(\large{ 125 }\) is the cube number.
• Cube Root  -  \(\large{ ^3\sqrt{125} = 5 }\) ,  \(\large{ 5 }\) is the cube root.  See nth Root.

D

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Decimal Number  -  Based on 10 digits.  \(\large{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }\)
• Decreasing Function  -  A function with a graph that moves downward as it is followed from left to right.
• Denominator  -  The number of equal parts of the whole is  \(\large{ 8 }\) , fraction is  \(\large{ \frac{3}{8} }\) .
• Digit  -  A numeral  \(\large{ 2119 }\)  has digits  \(\large{ 2, 1, 1, }\)  and  \(\large{ 9 }\) .
• Disjoint Event (Mutually Exclusive)  -  Events that have no outcomes in common.
• Distributive law of multiplication over addition  -  For any three numbers a, b, and c.  \(\large{a(b+c) = a (b)+a (c)}\), and \(\large{(b+c)(a) = b (a)+c (a)}\)
• Distributive law of multiplication over subtraction  -  For any three numbers a, b, and c.  \(\large{a(b−c) = a(b)−a(c)}\), and \(\large{(b−c)(a) = b (a)−c (a)}\)
• Distributive Postulate  -  Let \(\large{ a }\) , \(\large{ b }\) and \(\large{ c }\) be real numbers.  \(\large{\;a \left (b + c \right ) = ab + ac\; }\)
• Distributive Property (Distribution)  -  Multiply the parts of an expression  \(\large{ a \left(b-c \right) }\)  into another expression  \(\large{ a\;b-a\;c }\) .
• Dividend  -  In a set of terms  \(\large{ 3 \div 7 = 0.43 }\)  the amount to be divided.  \(\large{ 3 }\)  is the dividend,  \(\large{ 7 }\)  is the divisor, and  \(\large{ 0.43 }\)  is the quotient.
• Divisor  -  In a set of terms  \(\large{ 3 \div 7 = 0.43 }\)  the number divided by.  \(\large{ 7 }\)  is the divisor,  \(\large{ 3 }\)  is the dividend, and  \(\large{ 0.43 }\)  is the quotient.
• Division Postulate  -  A postulate is a statement that is assumed true without proof.  Let \(\large{ a, b, c }\) be real numbers.  If \(\large{ a=b }\) and \(\large{ c \ne 0 }\), then \(\large{ \frac{ a }{ c } = \frac{ b }{ c } }\) .
• Domain of a Function  -  A set of values for the independent variable that makes the function work.

E

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Element  -  Anything contained in a set.
• Elementary Algebra  -  Performs basic concepts of algebra operations.
  
• Engineering Notation  -  A way of writing large numbers  \(\large{ 1 2 3, 0 0 0 }\)  into smaller numbers  \(\large{ 1 2 3 \cdot 10^3 }\)  where the power of 10 is multiplied by 3.

• • \(\large{\; 1 2 3 4. 5  = 1. 2 3 4 5  \times 10^3}\)
  • \(\large{\; 1 2 0  = .1 2  \times 10^3}\)
  • \(\large{\; 1, 2 0 0 = 1. 2  \times 10^3}\)
  • \(\large{\; 1 2, 0 0 0 = 1 2 \times 10^3}\)
  • \(\large{\; 1 2 3, 0 0 0 = 1 2 3 \times 10^3}\)
  • \(\large{\; 1 2 3, 0 0 0, 0 0 0 = 1 2 3 \times 10^6}\)
  • \(\large{\; 1 2 3, 0 0 0, 0 0 0 = .1 2 3 \times 10^9}\)

• Equation  -  A statement containing one or more variables that are either added, subtracted, divided or multiplied to get an answer.  \(\large{ 13a^2+7x-21=19 }\)
• Elementary Arithmetic  -  Includes the simplified operations of addition, subtraction, division, and multiplication.
• Exponent (Index, Power)  -  Is how many times you multiply the number.  Term is \(\large{ 13a^2 }\), the exponent is \(\large{ 2 }\)
  
  • \(\large{ x^a + x^b = x^\left( a + b \right) }\)
  • \(\large{ x^a + y^a = \left( x \; y \right)^a }\)
  • \(\large{ \left( x^a \right)^b = x^\left( a \; b \right) }\)
  • \(\large{ x^{-a} = \frac{ 1 }{ x^a }  }\)
  • \(\large{ x^{a - b} = \frac{ x^a }{ x^b }  }\)

• Expression  -  A group of terms, coefficients, constants and variables separate by an operation.  An equation  \(\large{13a^2+7x-21=19 }\) , the expressions is  \(\large{ 13a^2+7x-21 }\)  and  \(\large{ 19 }\).

F

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Factor Number  -  Numbers \(\large{ 3 }\) and \(\large{ 8 }\) are factors that can be multiplied to get another number \(\large{ 24 }\) .  Equation \(\large{ 3 \times 8=24 }\)
• Factoring  -  Factor \(\large{ 7 \left(x-3\right) }\) expand to  \(\large{ 7x-21 }\)  or expressed as  \(\large{ 7 \left(x-3\right) = 7x-21 }\) .
• Factorial  -  The symbol is  \(\large{ ! }\) .  Multiply all whole numbers from the chosen number down to 1.  \(\large{ 5!=5\cdot 4\cdot 3\cdot 2\cdot 1=120 }\)  or  \(\large{ n!=\left(n+3\right) 2y\cdot 2\cdot 1=n }\)
• Formula  -  An expression in symbols used to calculate a desired result in mathematics and chemistry.
• Fractional Exponent  -  Is how mant times you multiply the number.  Term is \(\large{ 13a^{ \frac{2}{3} } }\), the exponent is \(\large{ \frac{2}{3} }\) .
• Fraction  -  A part  \(\large{ \frac{3}{8} }\)  of the whole.
  • Adding Fractions  -  \(\large{ \frac{a}{b}\;+\;\frac{c}{d} = \frac{ \left( a\;d \right) \;+\; \left( b\;c \right)  }{b\;d}   }\)
  • Subtract Fractions  -  \(\large{ \frac{a}{b}\;-\;\frac{c}{d} = \frac{ \left( a\;d \right) \;-\; \left( b\;c \right)  }{b\;d}   }\)
  • Multiply Fractions  -  \(\large{ \frac{a}{b}\;\frac{c}{d} = \frac{a\;c}{b\;d}   }\)
  • Divide Fractions  -  \(\large{ \frac{a}{b}\;\div\;\frac{c}{d} = \frac{ a\;d }{b\;c}   }\)

• Function  -  A relationship where a set of inputs (domain) determine a set of possible outputs (range).  The function of  \(\large{ f \left( x \right) = 5\;x }\)  is  \(\large{ f \left( x \right) }\) , the function name is  \(\large{ f }\) , the input value is  \(\large{ \left( x \right) }\) , and the output is (what the function does) \(\large{ 5\;x }\) .

G

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Geometric Mean  -  Two  numbers  \(\large{ a }\)  and  \(\large{ b }\)  is the number  \(\large{ c }\)  whose square equals the product  \(\large{ c^2 = a\;b }\) .
• Geometric Sequence (Geometric Progression)  -  Multipling the previous term by a constant.  \(\large{ 2 }\)  the sequence   \(\large{ 1, 2, 4, 8, 16, 32, ... }\)  or  \(\large{ b }\)  the sequence  \(\large{ a, ab, ab^2, ab^3, ... }\)
• Geometric Series  -  A series of the terms of a geometric sequence that has a constanr ratio.  \(\large{ 1 + 2 + 4 + 8 + 16 + 32 \;+ ... }\)
• Greatest Commom Factor  -  The highest number that divides exactly into two or more numbers.  factors of \(\large{12}\) are \(\large{ 1, 2, 3, 4, 6,12 }\) and factors of \(\large{16}\) are \(\large{ 1, 2, 4, 8, 16 }\), the greatest common factor of \(\large{12}\) and \(\large{16}\) is \(\large{4}\)

H

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Hexadecimal Number  -  Based on the number 16.  \(\large{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }\)

I

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Identity (Equation or Inequality)  -  An equation which is true regardless of what values are substituted for any variables (if there are any variables at all).  \(\large{ \left(a + b\right)^2 = a^2 + 2\;a\;b + b^2  }\)  or  \(\large{ 3 \cdot 7 = 21 }\)
• Imaginary Number  -  A number  \(\large{ i }\)  (imaginary symbol) when squared gives a negative number  \(\large{ i^2 = -1}\)  or  \(\large{\sqrt{-1} = i }\) .
  • Real  \(\large{ -2^2 = 4 }\)    Imaginary  \(\large{ 2i^2 = -4 }\)

• Improper Fraction  -  A fraction  \(\large{ \frac{21}{7} }\)  that has a larger numerator than denominator.
• Inequality  -  A mathematical sentence that uses one of the symbols
  • <   less than
  • >   more than
  • ≤   less than or equal to
  • ≥   more than or equal to

• Integer Number  -  A whole numbers that can be either positive or negative  \(\large{ ... , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }\)  with no fractions.
• Inverse (Reciprocal)  -  Reverses the effect of another number.  \(\large{ 3 \cdot 7 = 21 }\)  inverse is  \(\large{ \frac{21}{7}  = 3 }\) ,  \(\large{ 19 }\)  inverse is  \(\large{ -19 }\) .
• Irrational Number  -  A number that cannot be written as a fraction.  \(\large{ \sqrt{2} }\) ,  \(\large{ \pi=3.1415926535 ... }\) ,  \(\large{ e=2.71828182... }\)

J

K

L

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Like Terms  -  These are terms where the variables are the same.  The terms are  \(\large{ 13a^2, 3a^2, -3a^2 }\), the like terms are \(\large{ a^2 }\)  or the terms are  \(\large{ 13a^2 + 3a^2 + -3a^2 }\) , the like terms are  \(\large{ a^2 }\)
• Line  -  A straight path between two points or multiple points.
• Linear  -  In a straight line.

M

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Matrix  -  A rectangular or square array of numbers using either brackets  \(\large{ [\;] }\)  or parentheses  \(\large{ (\;) }\) .                   \({  \begin{bmatrix} 4 & 7 & 2.54 \\ -9 & 3.1 & 3 \\ 13 & 1.2 & -9 \end{bmatrix} }\)   or   \({ \begin{pmatrix} 4 & 7 & 2.54 \\ -9 & 3.1 & 3 & \\ 13 & 1.2 & -9 \end{pmatrix}  }\)
• Mean  -  The sum of all numbers in a set divided by the number of the values.  \(\large{ (2 + 3 + 4 + 5) / 4 = 3.5 }\)
• Mathematical Operation  -  addition \( \;(+),\; \) subtraction \(\;(-),\; \) multiplication \( \;(\times),\; \) division \( \;(+)\; \)
• Minuend  -  The first number in a set of terms  \(\large{ 3 - 7 = - 4 }\)  to be subtracted.  \(\large{ 3 }\)  is the minuend,  \(\large{ 7 }\)  is the subtrahend, and  \(\large{ -4 }\)  is the difference.
• Mixed Number  -  A number written as  \(\large{13 \frac{3}{8} }\)  a whole number  \(\large{13 }\) and a fraction  \(\large{ \frac{3}{8} }\) .
• Monomial  -  A polynomial with only one term  \(\large{ 13a^2 }\) .
• Mutually Exclusive (Disjoint Event)  -  Events that have no outcomes in common.
• Multiplicand  -  In a set of terms  \(\large{ 3 \times 7 = 21 }\)  the number that is multiplied.  \(\large{ 7 }\)  is the multiplicand,  \(\large{ 3 }\)  is the multiplier, and  \(\large{ 21 }\)  is the product.
• Multiplier  -  In a set of terms  \(\large{ 3 \times 7 = 21 }\)  the number that you are multiplying by.  \(\large{ 3 }\)  is the multiplier,  \(\large{ 7 }\)  is the multiplicand, and  \(\large{ 21 }\)  is the product.

N

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z                                              
• Natural number  -  Can be either counting numbers  \(\large{ 1, 2, 3, 4, 5, 6, ... }\)  or whole numbers  \(\large{ 0, 1, 2, 3, 4, 5, 6, ... }\) .
• Negative Exponent  -  Is how mant times you multiply the number.  Term is \(\large{ 13^{-2} = \frac{1}{13^2} = \frac{1}{169} }\), the exponent is \(\large{ -2 }\)
• Negative Number  -  It is the oposite of a whole number  \(\large{ ... , -5, -4, -3, -2, -1 }\)  or decimal number excluding  \(\large{ 0 }\) .
• nth Root  -  Some number  \(\large{ n }\) used as  \(\large{ ^n\sqrt{a} }\).        \(\large{ ^3\sqrt{27} = 27^{1/3} = 27^{0.33} = 3}\)       \(\large{ n^4 = n \cdot n \cdot n \cdot n }\)
• Number  -  A mathmatical object used to count.
• Number Line  -  Every point on a line represents a real number.
• Number Sentence  -  An equation of numbers and operations that expresses the relationship between them.  \(\large{ 3 + 7 = 10 \;,\; 3 < 7 }\)
• Number Properties  -  Associative, communitive, and distributive
• Number Types  - digits, fractional number, integer number, irrational number, natural number, numeral, rational number, real number, transcendental number, and whole number
• Numeral  -  A single symbol to make a numeral like  \(\large{ 2119 }\) .
• Numerator  -  The number of parts is  \(\large{ 3 }\), fraction is  \(\large{ \frac{3}{8} }\) .

O

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Octal Number  -  A number system based on 8  \(\large{ 0, 1, 2, 3, 4, 5, 6, 7 }\)
• Operator  -  A symbol such as  \(\large{ +, -, ... }\)
• Order of Operation  -  Parenthese (inside), exponents, multiplication and division (left to right), addition and subtraction (left to right) .  \(\large{ 13a^2 + 7x - 21 \left( 4 +1 \right) }\)
• Ordered Pair  -  Two numbers  \(\large{ \left(7, 21\right) }\)  or  \(\large{ \left(x, y\right) }\)  written in a certain order.
• Ordered Triple  -  Three numbers  \(\large{ \left(7, 21, 19\right) }\)  or  \(\large{ \left(x, y, z\right) }\)  written in a certain order.
• Ordered n  -  Multiple numbers  \(\large{ \left(7, 14, 21, ..., x_n\right) }\)  or  \(\large{ \left(x_1, x_2, x_3, ...,x_n\right) }\)  written in a certain order.

P

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Partial Fraction  -  A fraction  \(\large{\frac{3a^2 - 7x}{13a^2 + 7x - 21} }\)  that is broken into one or more smaller parts \(\large{\frac{a}{7x} + \frac{9}{4 + x}  }\) .
• Perfect Number  -  A whole number that is equal to the sum of its positive factors except the number itself.  \(\large{1+2+4+7=14}\) ,  \(\large{14}\) is a perfect number because the positive factors are  \(\large{1, 2, 4, 7,14}\) .
• Permutation  -  A set of objects in which the order is important.  \(\large{ \left(7, 21, 19\right) }\)
• Polynomial  -  The sum of two or more terms.  A term can have constants, exponents and variables, such as  \(\large{ 13a^2 }\) .  Put them together and you get a polynomial.
  • Monomial  -  1 term  \(\large{ 13a^2 }\)
  • Binomial  -  2 terms  \(\large{ 13a^2+7x }\)
  • Trinomial  -  3 terms  \(\large{ 13a^2+7x-21 }\)

• Porportional  -  When the ratio of two variables are constant.
• Positive Number  -  A counting number  \(\large{ 1, 2, 3, 4, 5, 6,... }\)  or decimal number excluding  \(\large{ 0 }\) .
• Postulate  -  A statement that is assumed true without proof.

• • Let \(\large{\;a}\) , \(\large{b}\) and \(\large{c}\) be real numbers.
  • • Reflexive Property  -  \(\large{a = a\; }\) (A quantity is congruent (equal) to itself.)
    • Symmetric Property  -  If \(\large{\;a = b\; }\), then \(\large{\;b = a }\)
    • Transitive Property  -  If \(\large{\;a = b\; }\) and \(\large{\;b = c\; }\) , then \(\large{\;a = c }\)
    • Addition Postulate  -  If \(\large{\;a = b\; }\) , then \(\large{\;a + c = b + c\; }\)
    • Subtraction Postulate  -  If \(\large{\;a = b\; }\) , then \(\large{\;a - c = b - c\; }\)
    • Multiplication Postulate  -  If \(\large{\;a = b\; }\) , then \(\large{\;ac = bc\; }\)
    • Division Postulate  -  If \(\large{\;a = b\; }\) and \(\large{\;c \ne 0\; }\) , then \(\large{\; \frac {a}{c} = \frac {b}{c}\; }\)
    • Substitution Postulate  -  If \(\large{\;a = b\; }\) , then \(\large{\;a\; }\) can be substituted for \(\large{\;b\; }\) in any expression.
    • Distributive Postulate  -  \(\large{\;a \left (b + c \right ) = ab + ac\; }\)
    • A straight line contains at least two points.
    • If two lines intersect, the intersection is only one point.
    • If two planes intersect, the intersection is only one line.
    • A plane must contain at least three noncollinear points.

• Power (Exponent, Index)  -  Is how mant times you multiply the number.  Term is \(\large{ 13a^2 }\), the exponent is \(\large{ 2 }\) .
• Prime Factor  -  A factor  \(\large{13, 7 }\)  are prime numbers.  \(\large{13\cdot 7 =91 }\)
• Prime Number  -  A number that can be divided evenly only by  \(\large{1}\) , or itself and it must be a whole number greater than \(\large{1}\) .
• Product  -  In a set of terms  \(\large{ 3 \times 7 = 21 }\)  the multiplied answer.  \(\large{ 21 }\)  is the product,  \(\large{ 3 }\)  is the multiplier, and  \(\large{ 7 }\)  is the multiplicand.  
• Proper Factor  -  Any of the factors of a number, except \(\large{1}\) or the number itself.
• Proper Fraction  -  When the numerator  \(\large{ 3 }\)  is less than the demominator  \(\large{ 8 }\)  of a fraction like  \(\large{ \frac{3}{8} }\) .

Q

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Quartile  -  One of three values that divide a data set into four equal sections.   \(\large{ 2, 4, 4, 5, 6, 7, 8 }\) , the quartiles are  \(\large{ 4 }\) (lower quartile), \(\large{ 5 }\) (middle quartile), and \(\large{ 7 }\) (upper quartile).
• Quotient  -  In a set of terms  \(\large{ 3 \div 7 = 0.43 }\)  the answer.  \(\large{ 0.43 }\)  is the quotient,  \(\large{ 3 }\)  is the dividend, and  \(\large{ 7 }\)  is the divisor.

R

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Radical  -  An expression  \(\large{ 13a^2+7x-23 }\)  that is a root  \(\large{ \sqrt{13a^2+7x-23} }\) .  The length of the bar  \(\large{ \sqrt{13a^2}+7x-23 }\)  tells how much of the expression is used.
• Radicand  -  The number under the symbol \(\large{ \sqrt{x} }\)
• Rational Number  -  Any number that can be expressed as a ratio (fraction) of two integers numbers.  \(\large{ 0=\frac{0}{1} }\) ,  \(\large{ 0.125=\frac{1}{8} }\) , \(\large{ 1.5=\frac{3}{2} }\)
• Real Number  -  Any number  \(\large{3, \frac{3}{4}, 13.45, -3.56, ... }\)  that is normally used.
• Reciprocal (Inverse)  -  Reverses the effect of another number.  \(\large{ 3\cdot 7 = 21 }\)  inverse is  \(\large{ \frac{21}{7}  = 3 }\) ,  \(\large{ 19 }\)  inverse is  \(\large{ -19 }\) .
• Remainder  -  What is left over after long division.  \(\large{ 7 \; / \;13 = 1 }\)  r \(\large{ 6 }\)
• Repeating Decimal  -  A decimal that keeps recurring over and over.  \(\large{ 0.\overline{33} }\)
• Rounding  -  Replacing a number  \(\large{ 3.1415926535 ... }\)  with another number having less digits  \(\large{ 3.1415 }\) .

S

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Scalar Number  -  Any single real number  \(\large{3, \frac{3}{4}, 13.45, -3.56, ... }\)  used to measure.
• Scientific Notation  -  A way of writing large numbers  \(\large{ 1 2 3 4 5 6 7 8 . 9 }\)  into two part  \(\large{ 1 2 3 4 5 . 6 7 8 9 \;x\; 10^3 }\) .
• Series  -  The sum of the terms of a sequence.  \(\large{ 1, 2, 3, 4, 5, 6, ... }\) or \(\large{ 1 + 2 + 3 + ... +\; n }\)
• Set  -  A group of numbers, variables, or really anything written using \(\large{ (\; ) }\) or \(\large{ [\; ] }\) .
• Sequence  -  A sequence of numbers in an orderly list.

• • \(\large{....,\; -15,\; -10,\; -5,\; 0,\; 5,\; 10,\; 15,\; ....}\)
  • \(\large{....,\; 1,\; 7,\; 14,\; 21,\; 28,\; 35,\; ....}\)
  • \(\large{....,\; -4.5,\; -3,\; -1.5,\; 0,\; 1.5,\; 3,\; 4.5,\; ....}\)

• Significant Digits  -  \(\large{ 1 2 3 0 }\)  Digits that are meaningful.  \(\large{ 0 . 0 1 2 3 0 }\)
• Simplifying  -  To simplify an algegtic expression  \(\large{ 5 \left( 2 + x \right) + 3 \left( 5x + 4 \right) - \left( x^2 \right)^2  }\)
  
  • Remove parentheses by multiplying factors.
  • Use exponent rules to remove parentheses in terms with exponents.
  • Combine like terms by adding coefficients.
  • Combine the constants.
    • \(\large{ = 5 \left( 2 + x \right) + 3 \left( 5x + 4 \right) - \left( x^2 \right)^2  }\)
    • \(\large{ = 10 + 5x + 15x + 12 - \left( x^2 \right)^2  }\)
    • \(\large{ = 10 + 5x + 15x + 12 - x^4  }\)
    • \(\large{ = 10 + 20x + 12 - x^4  }\)
    • \(\large{ = 22 + 20x - x^4  }\)
    • \(\large{ = - x^4 + 20x + 22  }\)

• Square Number  -  \(\large{ 5 \cdot 5 = 5^2 = 25 }\) ,  \(\large{ 25 }\) is the square number.
• Square Root  -  \(\large{ \sqrt{25} = 5 }\) ,  \(\large{ 5 }\) is the square root.
• Square Root Distribution  - 
  • \(\large{ \sqrt{ab} = \sqrt{a} \sqrt{b} }\)
  •                                                                                        
  • \(\large{ \sqrt{\frac{a}{b}} =  \frac{ \sqrt{a} }{ \sqrt{b} }  }\)        \(b \ne 0\)
  • \(\large{ ^n\sqrt{\frac{a}{b}} =  \frac{ ^n\sqrt{a} }{ ^n\sqrt{b} }  }\)        \(b \ne 0\)
  • \(\large{ \sqrt{a} \sqrt{a}  = a }\)
  • \(\large{ \sqrt{a^n} =  \left( \sqrt{a} \right)^n = a^{n/2}  }\)
  • Cannot
    • \(\large{ ^n\sqrt{ a + b } \ne \;  ^n\sqrt{ a } +\;  ^n\sqrt{ b }  }\)
    • \(\large{ ^n\sqrt{ a - b } \ne  \; ^n\sqrt{ a } - \; ^n\sqrt{ b }  }\)

• Standard Deviation  -  The square root of the variance.
• Subscript  -  A small letter or number lower than the normal text  \(\large{13_a^2 }\) .
• Subset  -  A  \(\large{\left( 3, 4, 5 \right) }\)  is a subset of B  \(\large{\left( 1, 2, 3, 4, 5, 6, 7, 8, 9 \right) }\) .
  • Empty Set - \(\large{ (\; ) }\)  is a  subset of B

• Subtrahend  -  In a set of terms  \(\large{ 3 - 7 = - 4 }\)  the number to be subtracted.  \(\large{ 7 }\)  is the subtrahend, \(\large{ 3 }\)  is the minuend, and  \(\large{ -4 }\)  is the difference.
• Sum  -  In a set of terms  \(\large{ 3 + 7 = 10 }\)  it is the result.  \(\large{ 10 }\) is the sum, and  \(\large{ 3 }\)  and  \(\large{ 7 }\)  are each addends.
• Superscript  -  A small letter or number higher than the normal text  \(\large{13_a^2 }\) .
• Surd  -  A square root  \(\large{\sqrt{2} }\)  that can not be simplified by removing the square root \(\large{\sqrt{2} }\) .  \(\large{\sqrt{4} }\) can be simplified to \(\large{2 }\) .
• Symmetry  -  Symmetry is when one shape becomes exactly like another if you flip or turn it.

T

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Terms  -  Either a single number, a variable, or numbers and variables.  An equation  \(\large{13a^2+7x-21=19 }\) , the terms are  \(\large{13a^2 }\) , \(\large{7x }\) , \(\large{21 }\) , and  \(\large{19 }\) .
• Theorem  -  A true statement that can be proven.

• • Congruence of Segments
    • • Segment congruence is reflexive, symmetric, and transitive.
        • • Reflexive - For any segment \(\;AB\; \), \(\;AB\; \)AB is congruent to \(\;AB\; \)
          • Symmetric - If \(\;AB = CD\; \) , then \(\;CD = AB\; \)
          • Transitive - If \(\;AB = CD\; \) and \(\;CD = EF\; \) . then \(\;AB = EF\; \)
  • Congruent Angles
    • Angle congruence is reflexive, symmetric, and transitive.
      • • Reflexive - For any \(\; \angle A\; \), \(\; \angle A\; = \angle A \)
        • Symmetric - If \(\; \angle A = \angle B \; \) , then \(\; \angle B = \angle A \; \)
        • Transitive - If \(\; \angle A = \angle B \; \) and \(\; \angle B = \angle C \; \), then \(\; \angle A = \angle C \; \)
  • Right Angle Congruence
    • All right angles are congruent.
  • Congruent Supplements
    • If two angles are supplementary to the same angle, then they are congruent.
    • If two angles are supplementary to congruent angles, then they are congruent.
  • Congruent Complementary
    • If two angles are complementary to the same angle, then they are congruent.
    • If two angles are complementary to congruent angles, then they are congruent.
  • Vertical Angles Congruence
    • Vertical angles are always congruent.

• Transcendental Number  -  A real number that cannot be the root of a polynomial equation with rational coefficients.  pi, e, Euler's constant, Catalan's constant, Liouville's number, Chaitin's constant, Chapernowne's number, Morse-Thue's number, Feigenbaum number
• Triangular Number  -  A number that can make a triangular dot pattern.  Each side in the triangle containes the same number of dots.  1 = 1 dot, 2 = 3 dots, 3 = 6 dots, 4 = 10 dots, 5 = 15 dots, etc.
• Trinomial  -  A polynomial with only three term  \(\large{ 13a^2+7x-21 }\) .

U

V

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Variable  -  Letters or symbols that are used to represent unknown values that can change depending in the infomation.  An equation  \(\large{13a^2+7x-21=19 }\) , the variables are  \(\large{a, x }\) .
• Vinculum  -  A line that is part of an expresson  \(\large{ \sqrt{a+b} }\)  or  \(\large{ \frac{a+b}{a-b} }\)  to show everything above or below the line is one group.

W

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Whole Number  -  Just positive numbers  \(\large{ 0, 1, 2, 3, 4, 5, 6, ... }\)  with no fractions.

X

Y

Z

• A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z
• Zero  -  A whole number that is neither  \(\large{ - }\)  or  \(\large{ + }\)  and contains no value.