Algebra

algebra banner 4Algebra 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. 

 

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Algebra Glossary

A

  • 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.
  • 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

  • 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

  • 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  \(\large{ 1 }\) .
  • 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  -  \(\large{ 5 \times 5 \times 5 = 125 }\) ,  \(\large{ 125 }\) 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.
  • 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.
  • Domain of a function  -  A set of values for the independent variable that makes the function work.

E

  • 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.
  • 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 mant times you multiply the number.  Term is \(\large{ 13a^2 }\), the exponent is \(\large{ 2 }\) .
  • 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

  • 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.
  • 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  \(\large{ 5\;x }\) .

G

  • 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.  \(\large{ 1 + 2 + 4 + 8 + 16 + 32 \;+ ... }\)

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.  \(\large{ 3\cdot 7 = 21 }\)  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

  • 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

  • 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.  \(\large{ (2 + 3 + 4 + 5) / 4 = 3.5 }\)
  • 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                                                   

  • 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  \(\large{ ^n\sqrt{a} }\).
  • 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

  • Octal number  -  \(\large{ 0, 1, 2, 3, 4, 5, 6, 7 }\)
  • Operator  -  A symbol such as  \(\large{ +, -, ... }\)
  • Order of operation  -  Parenthese (inside), exponents, multiplication and division (left to right), addition and subtraction (left to right)
  • 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

  • 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.
  • 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

  • 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

  • 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

  • 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  -  \(\large{ 5 \cdot 5 = 25 }\) ,  \(\large{ 25 }\) 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
  • 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 \(\large{2 }\) .

T

  • 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.
  • 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
  • Trinomial  -  A polynomial with only three term  \(\large{ 13a^2+7x-21 }\) .

U

V

  • 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

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

X

Y

Z

  • Zero  -  A whole number that is neither  \(\large{ - }\)  or  \(\large{ + }\)  and contains no value. 

 

 

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