## Algebra

Algebra is a branch of mathematics that uses letters or symbols as a place holder for unknown values or numbers.  These variables are used to represent relationships and to solve equations.  For other math terms see Geometry and Trigonometry.

## Algebra Terms

### A

• Absolute value - Makes a negative number positive  $$\large{ \left\vert -x \right\vert = x }$$  and positive numbers and  $$\large{ 0 }$$  are not changed.
• Axiom - A statement accepted as true without proof.

### B

• Base - The term  $$\large{13a^2 }$$  has a base  $$\large{ a }$$ .
• Binary numbers - Use only the digits $$\large{ 0 }$$ and $$\large{ 1 }$$ .
• Binomial - A polynomial with only two term  $$\large{ 13a^2+7x }$$ .

### C

• Coefficient - A number  $$\large{13a^2+7x-21=19 }$$  multiplied by a variable having the coefficients  $$\large{13, 7 }$$ .
• 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.
• 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  $$\large{ 1 }$$ .
• Conjugate - Is when you change the sign.  from  $$\large{ a+b }$$  to  $$\large{ a-b }$$,  from  $$\large{ 3a-4b }$$  to  $$\large{ 3a+4b }$$  $$\large{ ,... }$$
• 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 - $$\large{ 5 \;x\; 5 \;x\; 5 = 125 }$$ ,  $$\large{ 125 }$$ is the cube number.
• Cube root - $$\large{ ^3\sqrt{125} = 5 }$$ ,  $$\large{ 5 }$$ is the cube root.

### D

• Decimal number - Based on 10 digits.  $$\large{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }$$
• 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.
• Distribution - Multiply the parts of an expression  $$\large{ a \left(b-c \right) }$$  into another expression  $$\large{ a\;b-a\;c }$$ .
• Domain of a function - A set of values for the independent variable that makes the function work.

### E

• Equation - $$\large{ 13a^2+7x-21=19 }$$
• Exponent (also called index or power) - Is how mant times you multiply the number.  Term is $$\large{ 13a^2 }$$, the exponent is $$\large{ 2 }$$
• Expression - A group  $$\large{ 13a^2+7x-23 }$$  of terms, coefficients, constants and variables separate by an operation.

### F

• Factor number - Numbers $$\large{ 3 }$$ and $$\large{ 8 }$$ are factors that can be multiplied to get another number $$\large{ 24 }$$ .  Equation $$\large{ 3\;x\;8=24 }$$
• 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.  $$\large{ 5!=5\;x\;4\;x\;3\;x\;2\;x\;1=120 }$$  or  $$\large{ n!=\left(n+3\right) 2y\;x\;2\;x\;1=n }$$
• Fraction - A part  $$\large{ \frac{3}{8} }$$  of the whole.
• 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  $$\large{ 5\;x }$$ .

### H

• 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

• Imaginary number - A number  $$\large{ i }$$  (imaginary symbol) when squared gives a negative number  $$\large{ i^2 = -1}$$  or  $$\large{\sqrt{-1} = i }$$ .
• Improper fraction - A fraction  $$\large{ \frac{21}{7} }$$  that has a larger numerator than denominator.
• 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.  $$\large{ 3\;x\;7 = 21 }$$  inverse is  $$\large{ \frac{21}{7} = 3 }$$ ,  $$\large{ 19 }$$  inverse is  $$\large{ -19 }$$ .
• Irrational number - A number that cannot be written as a fraction.

### L

• 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  $$\large{ 13a^2 }$$

### M

• Matrix - A rectangular or square array of numbers using either brackets  $${ [\;] }$$  or parentheses  $${ (\;) }$$ .                                        $${ \begin{bmatrix} 4 & 7 & 2.54 \\ -9 & 3.1 & 3 \\ 13 & 1.2 & -9 \end{bmatrix} }$$   or   $${ \begin{pmatrix} 4 & 7 & 2.54 \\ -9 & 3.1 & 3 & \\ 13 & 1.2 & -9 \end{pmatrix} }$$
• Matrix addition - Matrices can be added term by term.                                                                                                                             $${ \begin{bmatrix} 4 & 7 & 2 \\ -9 & 3 & 3 \\ 13 & 1 & -9 \end{bmatrix} }$$ $${\;+\;}$$ $${ \begin{bmatrix} 5 & 3 & -2 \\ -4 & 6 & 2 \\ 11 & -1 & 14 \end{bmatrix} }$$ $${\;=\;}$$ $${ \begin{bmatrix} 9 & 10 & 0 \\ -13 & 9 & 5 \\ 24 & 0 & 5 \end{bmatrix} }$$
• Matrix subtraction - Matrices can be subtracted term by term.                                                                                                                 $${ \begin{bmatrix} 4 & 7 & 2 \\ -9 & 3 & 3 \\ 13 & 1 & -9 \end{bmatrix} }$$ $${\;-\;}$$ $${ \begin{bmatrix} 5 & 3 & -2 \\ -4 & 6 & 2 \\ 11 & -1 & 14 \end{bmatrix} }$$ $${\;=\;}$$ $${ \begin{bmatrix} -1 & 4 & 4 \\ -5 & -3 & 1 \\ 2 & 2 & -23 \end{bmatrix} }$$
• 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.

### N

• 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 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  $$\large{ ^n\sqrt{a} }$$ .
• Number line - Every point on a line represents a real 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

• Octal number -  $$\large{ 0, 1, 2, 3, 4, 5, 6, 7 }$$
• Operator - A symbol such as  $$\large{ +, -, ... }$$
• 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

• 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  $$\large{1, 2, 4, 7,14}$$ .
• 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  $$\large{ 0 }$$ .
• Prime factor - A factor  $$\large{13, 7 }$$  are prime numbers.  $$\large{13\;x\;7 =91 }$$
• 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 - When two or more values are multiplied togeather  $$\large{13a^2\;x\;7x \;x\;21=19 }$$ , the product is  $$\large{19 }$$ .
• Proper factor - Any of the factors of a number, except $$\large{1}$$ or the number itself.

### R

• 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 - Numbers expressed as a ratio of two numbers like  $$\large{ \frac{3}{4} }$$  or  $$\large{ -\frac{1}{8} }$$ .
• 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.  $$\large{ 3\;x\;7 = 21 }$$  inverse is  $$\large{ \frac{21}{7} = 3 }$$ ,  $$\large{ 19 }$$  inverse is  $$\large{ -19 }$$ .
• Rounding - Replacing a number  $$\large{ 3.1415926535 ... }$$  with another number having less digits  $$\large{ 3.1415 }$$ .

### S

• 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 $$\large{ [\; ] }$$ .
• Significant digits - $$\large{ 1 2 3 0 }$$  Digits that are meaningful.  $$\large{ 0 . 0 1 2 3 0 }$$
• Square number - $$\large{ 5 \;x\; 5 = 25 }$$ ,  $$\large{ 25 }$$ is the square number.
• Square root - $$\large{ \sqrt{25} = 5 }$$ ,  $$\large{ 5 }$$ is the square root.
• 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
• Superscript - A small letter or number higher than the normal text  $$\large{13_a^2 }$$ .

### T

• Terms - Either a single number, a variable, or numbers and variables  $$\large{ 13a^2, 7x, 21 }$$  of an expression  $$\large{ 13a^2+7x-23 }$$ .
• Theorem - A true statement that can be proven.
• Trinomial - A polynomial with only three term  $$\large{ 13a^2+7x-21 }$$ .

### V

• Variable - Letters or symbols that are used to represent unknown values that can change depending in the infomation.  The variables are $$\large{ a }$$ and $$\large{ x }$$

### W

• Whole number - Just positive numbers  $$\large{ 0, 1, 2, 3, 4, 5, 6, ... }$$  with no fractions.
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