Mathematics Page by Owen Borville February 12, 2026
Lesson 1: Number Systems: Sets of Real Numbers / Rules of Exponents - Scientific Notation / Scientific Notation (cont.) / Radicals Rational Exponents / Polynomials 1 / Polynomials II / Exponents - Polynomial Factoring
Real Numbers: are any numbers not imaginary= real numbers can be positive or negative, whole numbers or decimals are all real numbers.
Imaginary Numbers: a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by i^2=-1. The square of an imaginary number bi is =-b^2. EG=7i^2=-49
Complex Numbers: are numbers expressed by the form=a+bi, where a and b are real numbers and i is the imaginary unit, where i^2=-1.
Prime Numbers: Prime numbers are numbers that have only two factors: one and themself. Counting Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1
Integer: is a whole number, not decimal, including positive or negative or zero that can be added, subtracted, multiplied, or divided.
Rational Numbers: include both whole numbers, integers, or decimals
Irrational Numbers: cannot be expressed as a fraction, and have decimals that don't end or are not periodic.
Real Numbers: are any numbers not imaginary= real numbers can be positive or negative, whole numbers or decimals are all real numbers.
Imaginary Numbers: a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by i^2=-1. The square of an imaginary number bi is =-b^2. EG=7i^2=-49
Complex Numbers: are numbers expressed by the form=a+bi, where a and b are real numbers and i is the imaginary unit, where i^2=-1.
Prime Numbers: Prime numbers are numbers that have only two factors: one and themself. Counting Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1
Integer: is a whole number, not decimal, including positive or negative or zero that can be added, subtracted, multiplied, or divided.
Rational Numbers: include both whole numbers, integers, or decimals
Irrational Numbers: cannot be expressed as a fraction, and have decimals that don't end or are not periodic.
Arithmetic: Addition, Subtraction, Multiplication, Division, single digit, multiple digit, positive and negative numbers. Ex: 5+4, 5-4, 25+32, 25-32, (-4+2), 5x4, 25x32, (25/5), (-30/4)
Order of Operations in Mathematics: (1) Parentheses or brackets (2) Exponents (orders) (3) Multiplication and division (left to right) (4) Addition and subtraction (left to right)
Inequalities: Greater than/ Less than: Greater than: 5>4. Less than: 4<5. Equal: 5=5. Not equal: 5 is not 4. Greater than or equal, less than or equal. Comparing Integers.
Whole Numbers: have no decimal. Fractions: Adding, subtracting, multiplying, dividing fractions. Find common denominator to add or subtract fractions. Decimals: Add, subtract, multiply, divide: Ex: 5.4+2.3=7.7. A decimal is an integer and non-integer displayed with a “point” to describe a whole number and fraction value in numerical form= 56.25, 98.45, 6.5, 678.4
Percents: ⅕= 20 percent=20%. Percent is expressed symbolically and as a decimal. Ex: 5.2 percent=5.2%=0.052. Convert a fraction to percent.
Positive and Negative Numbers
Time Measurements: A.M., P.M. 1:00 Money Measurements: Counting money dollars, cents $1.50
Mixed Numbers: A mixed number is a whole number plus a proper fraction represented together. Ex: 6 ¼ is six and one fourth. 5 ⅕ is five and one fifth.
Ratio => Ex: Ratio of rock to sand=1:3 or 3:4=> one to three or three to four.
Rate is a value or quantity per unit time=> Ex: 20 miles per hour.
Proportion: a statement or equation with two equal ratios: (a/b)=(c/d) or a:b=c:d. Ex: (⅔)=(4/6), (5/15)=(⅓)
Percent: Ex: 54%=0.54. Pie Fractions whole, half, third, fourth, fifth, sixth,...etc. Percent: ⅕= 20 percent=20%. Percent is expressed symbolically and as a decimal. Ex: 5.2 percent=5.2%=0.052. Convert a fraction to percent
The Shapes of Geometry: can be in two dimensions or three dimensions
Square=four sides all equal length right angles, two dimensions
Rectangle=four sides, two equal, two not equal, right angles, two dimensions
Triangle=three sides, two dimensions
Circle=arch no sides, two dimensions
Cube=equal sides in three dimensions
Parallelogram=slanted rectangle or rectangle not with right angles, two dimensions
Sphere=three dimensional curved shape
Cone=three dimensional shape with one base,
Pyramid=three dimensional with four sided base,
Cylinder=curved two bases, three dimensions
Perimeter: length around object or area. Area: surface space inside object. Volume: three-dimensional space
Angles: 0 degrees to 360 degrees. Acute Angle: less than 90 degrees. Right Angle: 90 degrees. Obtuse Angle: greater than 90 degrees. Straight Angle: 180 degrees. Complete Angle: 360 degrees
Least Common Multiple LCM or LCF: the least common multiple of two integer numbers, a and b, is the smallest positive integer that is divisible by both numbers, a and b.
Ex: Multiples of 2=2, 4, 6, 8, 10, 12… Ex: Multiples of 3=3, 6, 9, 12, 15. LCM=6, 12. Factors of a mathematical value multiply together to equal the value.
Divisibility: in math, is a number divisible by another number? Divisibility test: for integers=EG=is the number divisible by 2,3,4,5,6,7,8,9,10,11,12,13,14,...etc.
Transformations: movement of position of objects in the coordinate plane graph. Four types of transformations: translation, rotation, reflection, and dilation.
Translation: moving object position on graph without changing size, shape, or orientation
Rotation: rotating object about a fixed point without changing size or shape
Dilation: expanding or contracting object without changing shape or orientation
Reflection: flipping object across a line without changing size or shape
Rigid versus Non-Rigid Transformation: rigid doesn't change shape or size=Non-rigid=changes size but not shape
Congruence in Geometry: Congruent geometry: objects with same shape and size (mirror image)
Greatest Common Factor (GCF): the largest common factor that two or more numbers share. Ex: 28:1, 2, 4, 7, 14, 28. Ex: 36:1, 2, 3, 4, 6, 9, 12, 18, 36
Factoring is breaking down a number or mathematical expression into its smallest multiplied parts or factors that can be multiplied together again to equal the original expression.
Factor Trees are a graphical representation of factoring a number or expression in a tree shape, where the original mathematical number or expression is listed first, then the factors are broken down below in a branching pattern until the smallest factors are shown.
Exponents: Exponents represent the number of times a number or math expression is multiplied by itself. B, the base number, to the x power is equal to B multiplied x times. Ex. 2^3=”two to the third power”=2x2x2. A number raised to the power of one is itself.
Exponent Rules=A and B are non-zero real numbers, m and n are any integers.
B^0=1
B^1=B
B^-n=(1/B^n)
(1/B^-n)=B^n
(B^m)(B^n)=B^m+n
(B^m)/(B^n)=B^m-n
(B^m)^n=B^mxn
(AB)^m=A^mxB^m
(A/B)^m=(A^m/B^m)
Radicals in mathematics indicate the root of a number, such as a square root, cube root, or nth root. Square Roots: Square Roots are factors of a number when multiplied equal the original number. Ex. Square Root of 49=7, or 7x7=49, Square Root of 81=9, or 9x9=81. Ex. Cube Root of 8=2, or 2x2x2=8, or 2^3=8.
Coordinate Plane: Algebra Formulas, Equations, Expressions, and Inequalities. One variable, two variables, ... etc. Ex: x+4=8. To find x, subtract each side by 4, then x=4.
Graphing Linear Equations and Inequalities: Absolute Value Equations and Graphs. Radical Expressions. Monomials. Polynomials
Scientific Notation: Scientific Notation is a simplified way of writing very large or small numbers by raising a number less than 10 by an exponent of base 10. The coefficient, base, and exponent make up the scientific notation. Ex: 35,000=3.5x10^4 => Coefficient=3.5 => Base=10 => Exponent=4 =>7.5x10^-4=0.00075
Significant Digits
Quadratic Formula: Quadratic Formula allows us to solve a quadratic equation. First step=convert the equation to form ax^2+bx+c=0 the x is unknown. Second step=Plug a,b,c into the formula=-b+-square root of (b^2-4ac)/(2a)
Parabola: Parabola is a symmetrical U-shaped curve that intersects with the vertical axis of symmetry at the vertex point where the parabola is most sharply curved. The x-intercepts are the same distance apart from the focus, a line from one leg of the parabola to the other, and also the vertex. A parabola represents a slice or plane from a three dimensional cone, also known as a conic section. Parabolas can be tall and skinny or short and wide. Parabolas can also be positioned above the x-axis or below the x-axis. Water streamed upward by a fountain forms a parabola shape, or any projectile motion or arch shape.
Pythagorean Theorem: States that in relation to the three sides of a right triangle, the sum of the squares of the leg lengths (opposite and adjacent) of the triangle are equal to the square of the hypotenuse or longest side of the triangle.
a^2+b^2=c^2
Topics in Mathematics: Definitions of number systems and sets, Properties of arithmetic operations, Linear equations in one variable, Solving absolute value equations, Graphing equations on the number line, The Cartesian plane, Graphing linear equations, Solving simultaneous linear equations, Exponents and powers, Roots and radicals, Polynomials, & Word problems
Equations: Inequalities, Units, Linear Equations: x and y intercept, Slope-Intercept form, point-slope form, Systems of Equations, Graphs, Functions, Sequences, arithmetic and geometric, Absolute Value functions and graphs, Exponents, Radicals, Exponential growth and decay, Quadratic polynomials, Factoring binomials and polynomials, Quadratic functions, equations, Irrational numbers.
Algebra Topics: Factoring Polynomials: Factoring, Division, Graphs, Rational exponents and radicals, Exponential models, Logarithms, Transformations of functions, Solving equations, Trigonometry, Rational functions, Polynomial basics, Quadratic equations in one variable, Inequalities in two variables, Graphing absolute value, Logarithms definition and laws, Sequences and series, Factorials, combinations, permutations, and Pascal's triangle, Probability, Complex numbers, Conic sections types and table.
Number Sequences: Arithmetic Sequence (progression by adding constant) in mathematics is a sequence of numbers where the difference (d) between each consecutive term is a constant so that=a, a+d, a+2d, a+3d.....etc. The nth term of an arithmetic sequence=an=a+(n-1)d. Ex.=4,9,14,19,24,29....etc, where the difference constant, d, is=5.
Geometric Sequence and Series: Geometric Sequence (progression by multiplying constant) =is an arrangement of numbers in a particular order where the next number comes after the previous number and is a sequence of numbers where each number after the first is found by multiplying the previous one by a fixed, non-zero number or common ratio= a, ar, ar^2, ar^3, ar^4...etc. The nth term of a geometric sequence =an=a1(r)^(n-1).
Geometric Series: is the sum of the terms of a geometric sequence with addition operations between them.
Harmonic Sequence: is the reciprocal of an INFINITE arithmetic progression sequence=an=1/[a1+(n-1)d] Ex: an=1/5, 1/10, 1/15, 1/20, 1/25, 1/30.....etc
Fibonacci Sequence (fibonacci numbers, Fn) is a series of numbers where a number is the addition of the LAST TWO NUMBERS, starting with 0 and 1. Applications= computer algorithms, many flowering plant petal patterns and tree branch patterns, leaf-stem patterns, and spiral patterns are arranged according to the fibonacci sequence. Ex: Artichoke flower, pineapple, banana plant.
Summation in math is the addition of a sequence of numbers that follow a pattern to find the sum. Summation is symbolized by the Greek letter Sigma (Σ). Notation for Sigma is=
n=upper limit (placed on top of Sigma)=end of sequence. x of k=argument (placed after Sigma) or xi, =element or expression or formula to determine the next value in the series. k=lower limit (placed below Sigma)=start of sequence.
Statistics: Mean is the sum of the number values divided by the amount of numbers in a numerical series. Median is when a set of numbers is ordered in ascending order and the middle numerical value is chosen. Mode is the most common number in a numerical series. Range is the difference between the highest number and the lowest number in a numerical series
Data: Standard Deviation in statistics is a measure of the amount of variation of a set of numerical values or how spread out they are. The symbol for standard deviation is the Greek letter sigma and the formula is the square root of the variance.
Variation: Variance Formula=How to find Variance: Find the mean value of a series. Subtract the mean value of a series from each number and square the result of each. Find the sum of these squares. Divide this by the number of observations minus one. Then find the square root of the total number.
Line Plots: Line Plots in statistics are graphs that show the frequency of data along a number line.
Histograms: Histogram in statistics show the approximate distribution of numerical data on a bar graph as columns on the x-axis.
Bar Charts: in statistics are a graphical representation of data as bars of different lengths based on values.
Box Plots: in statistics are graphical representations of data as boxes on a graph featuring their quartiles. Whiskers are lines extending from the boxes showing the variability of the values.
Probability in math is a numerical expression representing the likelihood of an event to occur. Probability values range from 0 to 1, where 0 is impossibility and 1 is certainty.
Factorials: in mathematics are the product of all positive integers less than or equal to a particular named integer, symbolized by that integer and an exclamation point. Example: Factorial of 7=7!=1x2x3x4x5x6x7. Factorial notation=n!=n(n-1)(n-2)(n-3)....etc. The graph of a factorial curves upward sharply.
0!=1
1!=1
2!=2x1=2
3!=3x2x1=6
4!=4x3x2x1=24
5!=5x4x3x2x1=120
6!=6x5x4x3x2x1=720
7!=7x6x5x4x3x2x1=5,040
Permutations versus Combinations in Mathematics (are often confused). Permutations (number of permutations) are arrangements of a series of numbers or objects in a particular order. Ex. (a,b) is different from (b,a). nPr=n!/(n-r)! n=set size (bigger number) r=subset size (smaller number) (n things taken r at a time and order matters). Ex: nPr=5P5=5!/(55)!=5x4x3x2x1/0!=120/1=120
Permutation: (order matters) eg. (lists) Versus Combination: (order doesn't matter) eg.(groups)
Combinations (Number of Combinations) in mathematics are the number of possible arrangements of a series of numbers or objects where the order of arrangement does not matter. nCr=n!/r!(n-r)! (n things taken r at a time and order doesn't matter). Ex: nCr=7C4=7!/(7-4)!4!=(7x6x5)/(3x2x1)=35=number of combinations
The Counting Principles: Product Rule involves the multiplication of events together to get a total number of outcomes of both events combined. The total number of outcomes of Event A and Event B combined is = AxB. Or, in other words, if there are A ways of doing one thing and B ways of doing another thing, then there are AxB ways of doing both things combined.
Sum Principle states that if there are A ways of doing one thing and B ways of doing another thing but both CANNOT be done together simultaneously, then the number of possible ways is A + B.
Simple Interest: A=P(1+rt)
A=final amount
P=original Principal amount
r=annual interest rate
t=time in years
Compounding Interest: in mathematics and finance is = A= P(1+r/n)^nt
A=final amount
P=original principal amount
r=rate of interest
t=time in years
n=number of times per year interest is compounded
Ex:
A=P(1+r/n)^nt
P=$1000
r=4%
t=2 yrs
n=10
A=1000(1+0.04/10)^10x2
Linear equations are equations between two variables that produce a straight line on a graph plot. Systems of Linear Equations are two or more linear equations combined together that can be graphically shown to intersect at a point on the graph.
Graphing Slope Intercept Form of Linear Equations
y=mx+b, of linear equations, is focused on the slope and y-intercept of the line.
y=y coordinate
m=slope
x=x coordinate
b=y intercept where slope intersects y axis
x-intercept=where the slope intersects the x axis
Slope Formula of a line by using x,y graphical coordinates: m=(y2-y1)/(x2-x1)
Point-Slope Form (formula) for linear equations: y2-y1=m(x2-x1)
Standard Form: Ax+By=C (where A, B, and C are integers)
Distance Formula: d=√ (square root of) [(x^2-x^1)^2+(y^2-y^1)^2]
Midpoint Formula: Find the midpoint on a graphical line by knowing the (x,y) coordinates of each end of the line. Ex: [(x1+x2)/2], [(y1+y2)/2]
Difference of Squares a binomial with perfect squares: a^2-b^2=(a+b)(a-b)
Factoring Polynomials and Algebraic Expressions: FOIL Method for factoring polynomials: First Outer Inner Last.
Greatest Common Factor method: Factor Trees. Ex: 2x^2-x-6 Factors: 2x,x Factors: +6,-6, +1, -1, +2, -2, +3, -3 =>(2x+3)(x-2)
Perfect Squares or Sum of Squares: (a+b)^2=(a+b)(a+b)=a^2+2ab+b^2
Difference of Squares: (a-b)^2=(a-b)(a-b)=a^2-2ab+b^2
Sum/Difference of Cubes: (a+b)^3 or (a-b)^3
Binomial is a two-term mathematical or algebraic expression
Trinomial is a three-term mathematical or algebraic expression
Polynomial is a multiple-term mathematical or algebraic expression
Linear Algebra and Matrices (Matrix)
Logarithms are the inverse function to exponentiation. A logarithm is the reverse of an exponent.
Natural Logs are logarithms with the base of e, the mathematical constant.
Venn Diagrams are illustrations that use overlapping circles to represent relationships, similarities, and differences among things or concepts where overlapping areas show similarities and non-overlapping areas show differences. Venn diagrams help show comparison, classification, groups, and categories between things.
Trigonometry deals with the geometry of right angled triangles to find angle measurements and side lengths. The ancient Babylonians and Egyptians developed trigonometry 2000 B.C.
Sine angle= opposite side length over hypotenuse=1/csc
Cosine angle =adjacent side length over hypotenuse=1/sec
Tangent angle=opposite side length over adjacent side=sin/cos
OH-AH-OA
Calculus was developed by Isaac Newton and Gottfried Leibniz, in the 17th century. Topics included: Functions, Limits, Derivatives (differentiation), and Integrals (integration). Bernard Riemann was influential in geometry, analysis, integration, differential geometry, and number theory. Differential equations were developed by Isaac Newton, Gottfried Leibniz, Jacob and Johan Bernoulli, and Leonhard Euler between the 17th century and 19th century.
Mathematics Number Timeline:
Babylonian math: (2000 B.C.) Sumerians created the first number system, which used a sexagesimal number system based on 60. Akkadians invented the abacus for counting numbers. Babylonians mastered trigonometry and quadratic equations. Babylonian clay tablets in Mesopotamia: Pythagorean theorem developed as early as 2000 B.C. along with geometry and arithmetic. Today’s time system is based on Babylonian system of 60 (seconds, minutes).
Egyptian Papyrus: (2000 B.C.) developed decimal numbers.
Indus Valley (2000 B.C.) developed decimal ratios.
Baudhayana (890 B.C.)
Greek Mathematics Euclid (700-200 B.C.)
Negative Numbers (100 B.C.) China
Roman Numerals.
Medieval Mathematics of Asia (Babylon, Persia, India, China)
Aryabhata (499 AD)
Brahmagupta (628 AD)first use of zero
Khwarizmi Algebra (820 AD) spread Arabic and Indian mathematics to Persia and to Europe.
Mayan Mexican mathematics developed the concept of zero independently.
Fibonacci Book: Liber Abaci (1202) Introduced Arabic number system to Europe.
Unique Numbers:
Pi
tau=2pi
Natural log base “e” Leonhard Euler
Imaginary number square of negative one
Imaginary number i to the i power
Riemann’s hypothesis=Apery’s constant
Prime numbers
Number 1
Euler’s identity
Euler’s number 2.718…
Number 0
Square root of 2=1.414…
Number 6174
Fibonacci sequence
Golden Ratio
Plank’s constant
Avogadro’s constant
Speed of light number
Infinity
Order of Operations in Mathematics: (1) Parentheses or brackets (2) Exponents (orders) (3) Multiplication and division (left to right) (4) Addition and subtraction (left to right)
Inequalities: Greater than/ Less than: Greater than: 5>4. Less than: 4<5. Equal: 5=5. Not equal: 5 is not 4. Greater than or equal, less than or equal. Comparing Integers.
Whole Numbers: have no decimal. Fractions: Adding, subtracting, multiplying, dividing fractions. Find common denominator to add or subtract fractions. Decimals: Add, subtract, multiply, divide: Ex: 5.4+2.3=7.7. A decimal is an integer and non-integer displayed with a “point” to describe a whole number and fraction value in numerical form= 56.25, 98.45, 6.5, 678.4
Percents: ⅕= 20 percent=20%. Percent is expressed symbolically and as a decimal. Ex: 5.2 percent=5.2%=0.052. Convert a fraction to percent.
Positive and Negative Numbers
Time Measurements: A.M., P.M. 1:00 Money Measurements: Counting money dollars, cents $1.50
Mixed Numbers: A mixed number is a whole number plus a proper fraction represented together. Ex: 6 ¼ is six and one fourth. 5 ⅕ is five and one fifth.
Ratio => Ex: Ratio of rock to sand=1:3 or 3:4=> one to three or three to four.
Rate is a value or quantity per unit time=> Ex: 20 miles per hour.
Proportion: a statement or equation with two equal ratios: (a/b)=(c/d) or a:b=c:d. Ex: (⅔)=(4/6), (5/15)=(⅓)
Percent: Ex: 54%=0.54. Pie Fractions whole, half, third, fourth, fifth, sixth,...etc. Percent: ⅕= 20 percent=20%. Percent is expressed symbolically and as a decimal. Ex: 5.2 percent=5.2%=0.052. Convert a fraction to percent
The Shapes of Geometry: can be in two dimensions or three dimensions
Square=four sides all equal length right angles, two dimensions
Rectangle=four sides, two equal, two not equal, right angles, two dimensions
Triangle=three sides, two dimensions
Circle=arch no sides, two dimensions
Cube=equal sides in three dimensions
Parallelogram=slanted rectangle or rectangle not with right angles, two dimensions
Sphere=three dimensional curved shape
Cone=three dimensional shape with one base,
Pyramid=three dimensional with four sided base,
Cylinder=curved two bases, three dimensions
Perimeter: length around object or area. Area: surface space inside object. Volume: three-dimensional space
Angles: 0 degrees to 360 degrees. Acute Angle: less than 90 degrees. Right Angle: 90 degrees. Obtuse Angle: greater than 90 degrees. Straight Angle: 180 degrees. Complete Angle: 360 degrees
Least Common Multiple LCM or LCF: the least common multiple of two integer numbers, a and b, is the smallest positive integer that is divisible by both numbers, a and b.
Ex: Multiples of 2=2, 4, 6, 8, 10, 12… Ex: Multiples of 3=3, 6, 9, 12, 15. LCM=6, 12. Factors of a mathematical value multiply together to equal the value.
Divisibility: in math, is a number divisible by another number? Divisibility test: for integers=EG=is the number divisible by 2,3,4,5,6,7,8,9,10,11,12,13,14,...etc.
Transformations: movement of position of objects in the coordinate plane graph. Four types of transformations: translation, rotation, reflection, and dilation.
Translation: moving object position on graph without changing size, shape, or orientation
Rotation: rotating object about a fixed point without changing size or shape
Dilation: expanding or contracting object without changing shape or orientation
Reflection: flipping object across a line without changing size or shape
Rigid versus Non-Rigid Transformation: rigid doesn't change shape or size=Non-rigid=changes size but not shape
Congruence in Geometry: Congruent geometry: objects with same shape and size (mirror image)
Greatest Common Factor (GCF): the largest common factor that two or more numbers share. Ex: 28:1, 2, 4, 7, 14, 28. Ex: 36:1, 2, 3, 4, 6, 9, 12, 18, 36
Factoring is breaking down a number or mathematical expression into its smallest multiplied parts or factors that can be multiplied together again to equal the original expression.
Factor Trees are a graphical representation of factoring a number or expression in a tree shape, where the original mathematical number or expression is listed first, then the factors are broken down below in a branching pattern until the smallest factors are shown.
Exponents: Exponents represent the number of times a number or math expression is multiplied by itself. B, the base number, to the x power is equal to B multiplied x times. Ex. 2^3=”two to the third power”=2x2x2. A number raised to the power of one is itself.
Exponent Rules=A and B are non-zero real numbers, m and n are any integers.
B^0=1
B^1=B
B^-n=(1/B^n)
(1/B^-n)=B^n
(B^m)(B^n)=B^m+n
(B^m)/(B^n)=B^m-n
(B^m)^n=B^mxn
(AB)^m=A^mxB^m
(A/B)^m=(A^m/B^m)
Radicals in mathematics indicate the root of a number, such as a square root, cube root, or nth root. Square Roots: Square Roots are factors of a number when multiplied equal the original number. Ex. Square Root of 49=7, or 7x7=49, Square Root of 81=9, or 9x9=81. Ex. Cube Root of 8=2, or 2x2x2=8, or 2^3=8.
Coordinate Plane: Algebra Formulas, Equations, Expressions, and Inequalities. One variable, two variables, ... etc. Ex: x+4=8. To find x, subtract each side by 4, then x=4.
Graphing Linear Equations and Inequalities: Absolute Value Equations and Graphs. Radical Expressions. Monomials. Polynomials
Scientific Notation: Scientific Notation is a simplified way of writing very large or small numbers by raising a number less than 10 by an exponent of base 10. The coefficient, base, and exponent make up the scientific notation. Ex: 35,000=3.5x10^4 => Coefficient=3.5 => Base=10 => Exponent=4 =>7.5x10^-4=0.00075
Significant Digits
Quadratic Formula: Quadratic Formula allows us to solve a quadratic equation. First step=convert the equation to form ax^2+bx+c=0 the x is unknown. Second step=Plug a,b,c into the formula=-b+-square root of (b^2-4ac)/(2a)
Parabola: Parabola is a symmetrical U-shaped curve that intersects with the vertical axis of symmetry at the vertex point where the parabola is most sharply curved. The x-intercepts are the same distance apart from the focus, a line from one leg of the parabola to the other, and also the vertex. A parabola represents a slice or plane from a three dimensional cone, also known as a conic section. Parabolas can be tall and skinny or short and wide. Parabolas can also be positioned above the x-axis or below the x-axis. Water streamed upward by a fountain forms a parabola shape, or any projectile motion or arch shape.
Pythagorean Theorem: States that in relation to the three sides of a right triangle, the sum of the squares of the leg lengths (opposite and adjacent) of the triangle are equal to the square of the hypotenuse or longest side of the triangle.
a^2+b^2=c^2
Topics in Mathematics: Definitions of number systems and sets, Properties of arithmetic operations, Linear equations in one variable, Solving absolute value equations, Graphing equations on the number line, The Cartesian plane, Graphing linear equations, Solving simultaneous linear equations, Exponents and powers, Roots and radicals, Polynomials, & Word problems
Equations: Inequalities, Units, Linear Equations: x and y intercept, Slope-Intercept form, point-slope form, Systems of Equations, Graphs, Functions, Sequences, arithmetic and geometric, Absolute Value functions and graphs, Exponents, Radicals, Exponential growth and decay, Quadratic polynomials, Factoring binomials and polynomials, Quadratic functions, equations, Irrational numbers.
Algebra Topics: Factoring Polynomials: Factoring, Division, Graphs, Rational exponents and radicals, Exponential models, Logarithms, Transformations of functions, Solving equations, Trigonometry, Rational functions, Polynomial basics, Quadratic equations in one variable, Inequalities in two variables, Graphing absolute value, Logarithms definition and laws, Sequences and series, Factorials, combinations, permutations, and Pascal's triangle, Probability, Complex numbers, Conic sections types and table.
Number Sequences: Arithmetic Sequence (progression by adding constant) in mathematics is a sequence of numbers where the difference (d) between each consecutive term is a constant so that=a, a+d, a+2d, a+3d.....etc. The nth term of an arithmetic sequence=an=a+(n-1)d. Ex.=4,9,14,19,24,29....etc, where the difference constant, d, is=5.
Geometric Sequence and Series: Geometric Sequence (progression by multiplying constant) =is an arrangement of numbers in a particular order where the next number comes after the previous number and is a sequence of numbers where each number after the first is found by multiplying the previous one by a fixed, non-zero number or common ratio= a, ar, ar^2, ar^3, ar^4...etc. The nth term of a geometric sequence =an=a1(r)^(n-1).
Geometric Series: is the sum of the terms of a geometric sequence with addition operations between them.
Harmonic Sequence: is the reciprocal of an INFINITE arithmetic progression sequence=an=1/[a1+(n-1)d] Ex: an=1/5, 1/10, 1/15, 1/20, 1/25, 1/30.....etc
Fibonacci Sequence (fibonacci numbers, Fn) is a series of numbers where a number is the addition of the LAST TWO NUMBERS, starting with 0 and 1. Applications= computer algorithms, many flowering plant petal patterns and tree branch patterns, leaf-stem patterns, and spiral patterns are arranged according to the fibonacci sequence. Ex: Artichoke flower, pineapple, banana plant.
Summation in math is the addition of a sequence of numbers that follow a pattern to find the sum. Summation is symbolized by the Greek letter Sigma (Σ). Notation for Sigma is=
n=upper limit (placed on top of Sigma)=end of sequence. x of k=argument (placed after Sigma) or xi, =element or expression or formula to determine the next value in the series. k=lower limit (placed below Sigma)=start of sequence.
Statistics: Mean is the sum of the number values divided by the amount of numbers in a numerical series. Median is when a set of numbers is ordered in ascending order and the middle numerical value is chosen. Mode is the most common number in a numerical series. Range is the difference between the highest number and the lowest number in a numerical series
Data: Standard Deviation in statistics is a measure of the amount of variation of a set of numerical values or how spread out they are. The symbol for standard deviation is the Greek letter sigma and the formula is the square root of the variance.
Variation: Variance Formula=How to find Variance: Find the mean value of a series. Subtract the mean value of a series from each number and square the result of each. Find the sum of these squares. Divide this by the number of observations minus one. Then find the square root of the total number.
Line Plots: Line Plots in statistics are graphs that show the frequency of data along a number line.
Histograms: Histogram in statistics show the approximate distribution of numerical data on a bar graph as columns on the x-axis.
Bar Charts: in statistics are a graphical representation of data as bars of different lengths based on values.
Box Plots: in statistics are graphical representations of data as boxes on a graph featuring their quartiles. Whiskers are lines extending from the boxes showing the variability of the values.
Probability in math is a numerical expression representing the likelihood of an event to occur. Probability values range from 0 to 1, where 0 is impossibility and 1 is certainty.
Factorials: in mathematics are the product of all positive integers less than or equal to a particular named integer, symbolized by that integer and an exclamation point. Example: Factorial of 7=7!=1x2x3x4x5x6x7. Factorial notation=n!=n(n-1)(n-2)(n-3)....etc. The graph of a factorial curves upward sharply.
0!=1
1!=1
2!=2x1=2
3!=3x2x1=6
4!=4x3x2x1=24
5!=5x4x3x2x1=120
6!=6x5x4x3x2x1=720
7!=7x6x5x4x3x2x1=5,040
Permutations versus Combinations in Mathematics (are often confused). Permutations (number of permutations) are arrangements of a series of numbers or objects in a particular order. Ex. (a,b) is different from (b,a). nPr=n!/(n-r)! n=set size (bigger number) r=subset size (smaller number) (n things taken r at a time and order matters). Ex: nPr=5P5=5!/(55)!=5x4x3x2x1/0!=120/1=120
Permutation: (order matters) eg. (lists) Versus Combination: (order doesn't matter) eg.(groups)
Combinations (Number of Combinations) in mathematics are the number of possible arrangements of a series of numbers or objects where the order of arrangement does not matter. nCr=n!/r!(n-r)! (n things taken r at a time and order doesn't matter). Ex: nCr=7C4=7!/(7-4)!4!=(7x6x5)/(3x2x1)=35=number of combinations
The Counting Principles: Product Rule involves the multiplication of events together to get a total number of outcomes of both events combined. The total number of outcomes of Event A and Event B combined is = AxB. Or, in other words, if there are A ways of doing one thing and B ways of doing another thing, then there are AxB ways of doing both things combined.
Sum Principle states that if there are A ways of doing one thing and B ways of doing another thing but both CANNOT be done together simultaneously, then the number of possible ways is A + B.
Simple Interest: A=P(1+rt)
A=final amount
P=original Principal amount
r=annual interest rate
t=time in years
Compounding Interest: in mathematics and finance is = A= P(1+r/n)^nt
A=final amount
P=original principal amount
r=rate of interest
t=time in years
n=number of times per year interest is compounded
Ex:
A=P(1+r/n)^nt
P=$1000
r=4%
t=2 yrs
n=10
A=1000(1+0.04/10)^10x2
Linear equations are equations between two variables that produce a straight line on a graph plot. Systems of Linear Equations are two or more linear equations combined together that can be graphically shown to intersect at a point on the graph.
Graphing Slope Intercept Form of Linear Equations
y=mx+b, of linear equations, is focused on the slope and y-intercept of the line.
y=y coordinate
m=slope
x=x coordinate
b=y intercept where slope intersects y axis
x-intercept=where the slope intersects the x axis
Slope Formula of a line by using x,y graphical coordinates: m=(y2-y1)/(x2-x1)
Point-Slope Form (formula) for linear equations: y2-y1=m(x2-x1)
Standard Form: Ax+By=C (where A, B, and C are integers)
Distance Formula: d=√ (square root of) [(x^2-x^1)^2+(y^2-y^1)^2]
Midpoint Formula: Find the midpoint on a graphical line by knowing the (x,y) coordinates of each end of the line. Ex: [(x1+x2)/2], [(y1+y2)/2]
Difference of Squares a binomial with perfect squares: a^2-b^2=(a+b)(a-b)
Factoring Polynomials and Algebraic Expressions: FOIL Method for factoring polynomials: First Outer Inner Last.
Greatest Common Factor method: Factor Trees. Ex: 2x^2-x-6 Factors: 2x,x Factors: +6,-6, +1, -1, +2, -2, +3, -3 =>(2x+3)(x-2)
Perfect Squares or Sum of Squares: (a+b)^2=(a+b)(a+b)=a^2+2ab+b^2
Difference of Squares: (a-b)^2=(a-b)(a-b)=a^2-2ab+b^2
Sum/Difference of Cubes: (a+b)^3 or (a-b)^3
Binomial is a two-term mathematical or algebraic expression
Trinomial is a three-term mathematical or algebraic expression
Polynomial is a multiple-term mathematical or algebraic expression
Linear Algebra and Matrices (Matrix)
Logarithms are the inverse function to exponentiation. A logarithm is the reverse of an exponent.
Natural Logs are logarithms with the base of e, the mathematical constant.
Venn Diagrams are illustrations that use overlapping circles to represent relationships, similarities, and differences among things or concepts where overlapping areas show similarities and non-overlapping areas show differences. Venn diagrams help show comparison, classification, groups, and categories between things.
Trigonometry deals with the geometry of right angled triangles to find angle measurements and side lengths. The ancient Babylonians and Egyptians developed trigonometry 2000 B.C.
Sine angle= opposite side length over hypotenuse=1/csc
Cosine angle =adjacent side length over hypotenuse=1/sec
Tangent angle=opposite side length over adjacent side=sin/cos
OH-AH-OA
Calculus was developed by Isaac Newton and Gottfried Leibniz, in the 17th century. Topics included: Functions, Limits, Derivatives (differentiation), and Integrals (integration). Bernard Riemann was influential in geometry, analysis, integration, differential geometry, and number theory. Differential equations were developed by Isaac Newton, Gottfried Leibniz, Jacob and Johan Bernoulli, and Leonhard Euler between the 17th century and 19th century.
Mathematics Number Timeline:
Babylonian math: (2000 B.C.) Sumerians created the first number system, which used a sexagesimal number system based on 60. Akkadians invented the abacus for counting numbers. Babylonians mastered trigonometry and quadratic equations. Babylonian clay tablets in Mesopotamia: Pythagorean theorem developed as early as 2000 B.C. along with geometry and arithmetic. Today’s time system is based on Babylonian system of 60 (seconds, minutes).
Egyptian Papyrus: (2000 B.C.) developed decimal numbers.
Indus Valley (2000 B.C.) developed decimal ratios.
Baudhayana (890 B.C.)
Greek Mathematics Euclid (700-200 B.C.)
Negative Numbers (100 B.C.) China
Roman Numerals.
Medieval Mathematics of Asia (Babylon, Persia, India, China)
Aryabhata (499 AD)
Brahmagupta (628 AD)first use of zero
Khwarizmi Algebra (820 AD) spread Arabic and Indian mathematics to Persia and to Europe.
Mayan Mexican mathematics developed the concept of zero independently.
Fibonacci Book: Liber Abaci (1202) Introduced Arabic number system to Europe.
Unique Numbers:
Pi
tau=2pi
Natural log base “e” Leonhard Euler
Imaginary number square of negative one
Imaginary number i to the i power
Riemann’s hypothesis=Apery’s constant
Prime numbers
Number 1
Euler’s identity
Euler’s number 2.718…
Number 0
Square root of 2=1.414…
Number 6174
Fibonacci sequence
Golden Ratio
Plank’s constant
Avogadro’s constant
Speed of light number
Infinity