by Owen Borville

**January 19, 2023**

Survey of Mathematics

Number Systems

Counting Numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Arithmetic: Addition, Subtraction, Multiplication, Division

single digit, multiple digit, positive and negative numbers

5+4, 5-4, 25+32, 25-32, (-4+2), 5x4, 25x32, (25/5), (-30/4)

Order of Operations

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

No decimal

**Fractions**

Adding, subtracting, multiplying, dividing fractions

Decimals

Add, subtract, multiply, divide

5.4+2.3=7.7

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

Percents 54%=0.54

Pie Fractions whole, half, third, fourth, fifth, sixth,...etc

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

6 ¼ is six and one fourth. 5 ⅕ is five and one fifth

Ratios=EG.=Ratio of rock to sand=1:3 or 3:4=one to three or three to four

Rates=a value or quantity per unit time=EG.=20 miles per hour

Proportions= a statement or equation with two equal ratios

(a/b)=(c/d) or a:b=c:d

(⅔)=(4/6), (5/15)=(⅓)

**Percents**

Percents=⅕= 20 percent=20%

Percent is expressed symbolically and as a decimal

5.2 percent=5.2%=0.052

Convert a fraction to percent

Positive and Negative Numbers

Real Numbers

Real Numbers are any numbers not imaginary= real numbers can be positive or negative, whole numbers or decimals are all real numbers.

**Imaginary 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

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 Prime numbers are numbers that have only two factors: one and themself.

Shapes of Geometry

Two Dimensions Three Dimensions

Square, Rectangle, Triangle, Circle, Cube, Parallelogram, Sphere, Cone, Pyramid, Cylinder

Perimeter, Area, Volume

Square=four sides all equal length right angles

Rectangle=four sides, two equal, two not equal, right angles,

Triangle=three sides,

Circle=arch no sides,

Cube=equal sides in three dimensions,

Parallelogram=slanted rectangle or rectangle not with right angles,

Sphere=three dimensional curved shape,

Cone=three dimensional shape with one base,

Pyramid=three dimensional with four sided base,

Cylinder=curved two bases

Perimeter=length

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.

Multiples of 2=2, 4, 6, 8, 10, 12…

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.

Integer=is a whole number, not decimal, including positive or negative or zero

Add Subtract Multiply Divide

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

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

28:1, 2, 4, 7, 14, 28

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)

**Square Roots**

Square Roots are factors of a number when multiplied equal the original number

Square Root of 49=7, or 7x7=49, Square Root of 81=9, or 9x9=81.

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

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.

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

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 2 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

EG.=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]

EG.=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. EG. Artichoke flower, pineapple, banana plant.

**Summation**

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. 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**

Bar Charts in statistics are a graphical representation of data as bars of different lengths based on values.

Box Plots

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

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**

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 (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)

EG.

nPr=5P5=5!/(5-5)!=5x4x3x2x1/0!=120/1=120

Permutation (order matters) eg. (lists) VS

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)

EG.

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

EG.

A=P(1+r/n)^nt

P=$1000

r=4%

t=2 yrs

n=10

A=1000(1+0.04/10)^10x2

Linear Equations

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.

[(x1+x2)/2], [(y1+y2)/2]

**Difference of Squares**a binomial with perfect squares

a^2-b^2=(a+b)(a-b)

Factoring Polynomials

Algebraic Expressions

FOIL Method for factoring polynomials

First Outer Inner Last

Greatest Common Factor method

Factor Trees

EG.

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 Lebiniz, in the 17th century.

Functions

Limits

Derivatives

Integrals

Bernard Riemann

Differential Equations

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 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.)

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.

Negative Numbers (100 B.C.) China

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