General Mathematics - Number Sense, Patterns, and Algebraic Thinking

a + (b + c)

(a + b) + c =

Associative Property of Addition:

(a + b) + c = a + (b + c)

Associative Property of Multiplication:

(a · b) · c = a · (b · c)

a × (b × c)

(a × b) × c =

Division is not Associative

(a ÷ b) ÷ c does not have to be the same as a ÷ (b ÷ c).

Euclid's Postulates

1. Any two points can be joined by a straight line. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Least Common Multiple (LCM)

1. Find the prime factorizations of each number. • To find the prime factorization one method is a factor tree where you begin with any two factors and proceed by dividing the numbers until all the ends are prime factors. 2. Star factors which are shared. 3. Then multiply the un-starred factors of one of the numbers by the other number. (This is the LCD)

Greatest Common Factor (GCF)

1. Find the prime factorizations of each number. • To find the prime factorization one method is a factor tree where you begin with any two factors and proceed dividing the numbers until you reach all prime factors. 2. Star factors which are shared. 3. Then multiply the starred factors of either number. (This is the GCF)

Order of Operations - PEMDAS "Please Excuse My Dear Aunt Sally"

1. Parentheses (or any grouping symbol {braces}, [square brackets], |absolute value|) • Complete all operations inside the parentheses first following the order of operations 2. Exponents 3. Multiplication & Division from left to right 4. Addition & Subtraction from left to right

4 + x = 12

4 more than a certain number is 12

Countable

A "countable" infinite set is one that can be put into one-to-one correspondence with the set of natural numbers.

Configuration Space

A configuration space is a topological object that can be used to study the allowable states of a given system.

prime factors

A factor tree is a way to visualize a number's

Complete Graph

A graph in which every node is connected to every other node is called a complete graph.

Group

A group is just a collection of objects (i.e., elements in a set) that obey a few rules when combined or composed by an operation. In order for a set to be considered a group under a certain operation, each element must have an inverse, the set must have an identity element, the set must be closed, and the set must be associative under the operation.

if it is an even number (the last digit is 0, 2, 4, 6 or 8)

A number is divisible by 2

Hyperland

A point in four-space, also known as 4-D space, requires four numbers to fix its position. Four-space has a fourth independent direction, described by "ana" and "kata."

Line Land

A point in one dimension requires only one number to define it. The number line is a good example of a one-dimensional space.

Spaceland

A point in three-dimensional space requires three numbers to fix its location.

Polynomial

A polynomial is an algebraic "sentence" containing an unknown quantity.

Hypersphere

A sphere can be thought of as a stack of circular discs of increasing, then decreasing, radii. The process of slicing is one way to visualize higher-dimensional objects via level curves and surfaces. A hypersphere can be thought of as a "stack" of spheres of increasing, then decreasing, radii.

Prime Number

A whole number (other than 1) is a prime number if its only factors (divisors) are 1 and itself. Equivalently, a number is prime if and only if it has exactly two factors (divisors).

Central Limit Theorem

According to the Central Limit Theorem, the distribution of averages of many trials is always normal, even if the distribution of each trial is not.

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Add and subtract

does not change the solution set.

Adding the same quantity to both sides of an equation if a = b, then adding c to both sides of the equation produces the equivalent equation a + c = b + c.

1. The unit 2. Prime numbers 3. Composite numbers

All integers are thus divided into three classes:

Permutation

An arrangement where order matters.

variable

An important part of problem solving is identifying

Tone

An instrument's tone, the sound it produces, is a complex mixture of waves of different frequencies.

Continuous Symmetry

An object possessing continuous symmetries can remain invariant while one symmetry is turned into another. A circle is an example of an object with continuous symmetries.

Rarefactior

Assuming that the air is of uniform density and pressure to begin with, a region of high pressure will be balanced by a region of low pressure, called rarefaction, immediately following the compression

Factor Trees

At each level of the tree, break the current number into a product of two factors. The process is complete when all of the "circled leaves" at the bottom of the tree are prime numbers. Arranging the factors in the "circled leaves" in order. The final answer does not depend on product choices made at each level of the tree.

The Same

Because of the associate property of addition, when presented with a sum of three numbers, whether you start by adding the first two numbers or the last two numbers, the resulting sum is

Box Diagram

Box diagrams, also known as gluing diagrams, are a convenient way to examine intrinsic topology.

Aleph-Null

Cantor called the cardinality of all the sets that can be put into one-to-one correspondence with the counting numbers , or "Aleph Null."

Conditional Probability

Conditional probability determines the likelihood of events that are not independent of one another.

Dimension

Dimension is how mathematicians express the idea of degrees of freedom—aspects of an object that can be measured separately.

Division by Zero

Division by zero is undefined. Each of the expressions 6 ÷0 and 6/0 and 0)6 is undefined.

General Relativity

Einstein's famous theory of General Relativity relates gravity to the curvature of spacetime.

Set up an Equation

Every solution to a word problem must include a carefully crafted equation that accurately describes the constraints in the problem statement.

Unique Factorization Theorem

Every whole number can be uniquely factored as a product of primes. This result guarantees that if the prime factors are ordered from smallest to largest, everyone will get the same result when breaking a number into a product of prime factors.

Extrinsic View

Extrinsic topology is the study of shape from an external perspective.

Fourier Analysis and Synthesis

Fourier analysis breaks a complicated signal into a combination of simple sine waves. Fourier synthesis does the opposite, constructing a complicated signal from simple sine waves.

Fourier Analysis

Fourier analysis is the process of taking a complicated signal and breaking it into sine and cosine components.

Geometry

Geometry is the mathematical study of space. The geometry of a space goes hand in hand with how one defines the shortest distance between two points in that space.

a + c = b + c

If a = b then

a · c = b · c for c does not equal 0

If a = b then

a ÷ c = b ÷ c for c does not equal 0

If a = b then

a − c = b − c

If a = b then

Properties of Equality

If a = b then a + c = b + c If a = b then a − c = b − c If a = b then a · c = b · c for c does not equal 0 If a = b then a ÷ c = b ÷ c for c does not equal 0

Commutative Property of Multiplication

If a and b are any whole numbers, then a · b = b · a.

The Multiplicative Identity Property

If a is any whole number, then a ·1 = a and 1 · a = a.

Multiplication by Zero

If a represents any whole number, then a · 0 = 0 and 0 · a = 0.

Composite Numbers

If a whole number is not a prime number, then it is called a composite number.

The Associative Property of Multiplication

If a, b, and c are any whole numbers, then a · (b · c) = (a · b) · c.

evaluate the expression in the innermost pair of grouping symbols first.

If grouping symbols are nested

Non-Orientability

If on a surface there is no meaningful way to tell an object's orientation (left or right handedness), the surface is said to be non-orientable.

The inverse of addition is subtraction

If we start with a number x and add a number a, then subtracting a from the result will return us to the original number x. x + a − a = x. so,

The inverse of multiplication is division

If we start with a number x and multiply by a number a, then dividing the result by the number a returns us to the original number x. In symbols, a · x / a = x.

The inverse of subtraction is addition

If we start with a number x and subtract a number a, then adding a to the result will return us to the original number x. In symbols, x − a + a = x. So,

Denominator

In any ratio of two whole numbers, expressed as a fraction, we can interpret the first (top) number to be the "counter," or numerator—that which indicates how many pieces—and the second (bottom) number to be the "namer," or denominator—that which indicates the size of each piece.

Continuous

In some ways, the opposite of a multitude is a magnitude, which is continuous. In other words, there are no well defined partitions.

Products and Factors

In the expression 3 · 4, the whole numbers 3 and 4 are called the factors and 3 · 4 is called the product.

Hyperbolic Geometry

In this type of geometry the angles of a triangle add up to less than 180 degrees. In such a system, one has to replace the parallel postulate with a version that admits many parallel lines.

Spherical Geometry

In this type of geometry the angles of a triangle add up to more than 180 degrees. In such a system, one has to replace the parallel postulate with a version that admits no parallel lines as well as modify Euclid's first two postulates.

Overtone

Instruments produce notes that have a fundamental frequency in combination with multiples of that frequency known as partials or overtones

Intrinsic View

Intrinsic topology is the study of shape from the perspective of being on the surface of the shape.

Irrational

Irrational numbers cannot be written as a ratio of natural numbers.

Look Back

It is important to note that this step does not imply that you should simply check your solution in your equation. After all, it's possible that your equation incorrectly models the problem's situation, so you could have a valid solution to an incorrect equation. The important question is: "Does your answer make sense based on the words in the original problem statement."

Divisible

Let a and b be whole numbers. Then a is divisible by b if and only if the remainder is zero when a is divided by b. In this case, we say that "b is a divisor of a."

The Commutative Property of Addition

Let a and b represent two whole numbers. Then, a + b = b + a.

The Distributive Property (Subtraction)

Let a, b, and c be any whole numbers. Then, a · (b − c) = a · b − a · c. We say the multiplication is "distributive with respect to subtraction."

Associate Property of Addition

Let a, b, and c represent whole numbers. Then, (a + b) + c = a + (b + c).

Markov Chains

Markov chains are a way to analyze sequences of events where the outcomes of prior events affect the probability of outcomes of subsequent events.

repeated addition

Multiplication is equivalent to

the set of natural numbers

N = {1, 2, 3, 4, 5, . . .}.

Non-Euclidian Geometry

Non-Euclidean geometries abide by some, but not all of Euclid's five postulates.

Irrational

Objects are topologically equivalent if they can be continuously deformed into one another. Properties that are preserved during this process are called topological invariants.

Pigeonhole Principle

Of central importance in Ramsey Theory, and in combinatorics in general, is the "pigeonhole principle," also known as Dirichlet's box. This principle simply states that we cannot fit n+1 pigeons into n pigeonholes in such a way that only one pigeon is placed in each hole, with no pigeons left over.

B − 125 = 1200

Original Balance minus Amelie's Withdrawal is Current Balance

left to right

Perform all additions and subtractions in the order presented

Flat Land

Points in two-dimensional space require two numbers to specify them completely. The Cartesian plane is a good way to envision two-dimensional space.

Principal Curvatures

Principal curvatures are a way to extrinsically measure the curvature of a surface by looking at a given point and finding the contour line with the greatest curvature and the contour line with the least curvature.

Probability

Probability theory enables us to use mathematics to characterize and predict the behavior of random events. By "random" we mean "unpredictable" in the sense that in a given specific situation, our knowledge of current conditions gives us no way to say what will happen next.

Public Key Encryption

Public key encryption allows two parties to communicate securely over an un-secured computer network using the properties of prime numbers and modular arithmetic. RSA is the modern standard for public key encryption.

Ramsey Theory

Ramsey Theory reveals why we tend to find structure in seemingly random sets. Ramsey numbers indicate how big a set must be to guarantee the existence of certain minimal structures.

Rational

Rational numbers arise from the attempt to measure all quantities with a common unit of measure.

1. Set up a Variable Dictionary. 3. Solve the Equation. 4. Answer the Question. 5. Look Back.

Requirements for Word Problem Solutions.

1. Set up a Variable Dictionary. You must let your readers know what each variable in your problem represents. This can be accomplished in a number of ways: • Statements such as "Let P represent the perimeter of the rectangle." • Labeling unknown values with variables in a table. • Labeling unknown quantities in a sketch or diagram. 2. Set up an Equation. Every solution to a word problem must include a carefully crafted equation that accurately describes the constraints in the problem statement. 3. Solve the Equation. You must always solve the equation set up in the previous step. 4. Answer the Question. This step is easily overlooked. For example, the problem might ask for Jane's age, but your equation's solution gives the age of Jane's sister Liz. Make sure you answer the original question asked in the problem. Your solution should be written in a sentence with appropriate units. 5. Look Back. It is important to note that this step does not imply that you should simply check your solution in your equation. After all, it's possible that your equation incorrectly models the problem's situation, so you could have a valid solution to an incorrect equation. The important question is: "Does your answer make sense based on the words in the original problem statement."

Requirements for Word Problem Solutions.

1. Mark the place you wish to round to. This is called the rounding digit . 2. Check the next digit to the right of your digit marked in step 1. This is called the test digit . If the test digit is greater than or equal to 5, add 1 to the rounding digit and replace all digits to the right of the rounding digit with zeros. If the test digit is less than 5, keep the rounding digit the same and replace all digits to the right of rounding digit with zeros.

Rules for Rounding - To round a number to a particular place, follow these steps:

...

Since a fraction is just a division we can perform a division and get a representation of the number.

1. Simplify the expression on either side of the equation. 2. Gather the variable term on the left-hand side (LHS) by adding to both sides. the opposite of the variable term on the right-hand side (RHS). • Note: either side is fine but we will consistently place the variable on the LHS. when a variable is present on both sides of the equation 3. Gather the constant terms on the RHS by adding to both sides the opposite of the constant term on the LHS. 4. Divide both sides by the coefficient of the variable 5. Simplify if possible. 6. Check your solution.

Solving Equations

Factor Tree Alternate Approach

Some favor repeatedly dividing by 2 until the result is no longer divisible by 2. Then try repeatedly dividing by the next prime until the result is no longer divisible by that prime. The process terminates when the last resulting quotient is equal to the number 1.

Figurate Numbers

Some numbers make geometric shapes when arranged as a collection of dots, for example, 16 makes a square, and 10 makes a triangle.

Symmetry

Symmetry, in a mathematical sense, is a transformation that leaves an object invariant. Symmetry is perhaps most familiar as an artistic or aesthetic concept. Designs are said to be symmetric if they exhibit specific kinds of balance, repetition, and/or harmony. In mathematics, symmetry is more akin to something like "constancy," that is, how something can be manipulated without changing its form.

Euler Characteristic

The Euler characteristic is a topological invariant that relates a surface's vertices, edges, and faces.

Galton Board

The Galton board is a model of a sequence of random events. Each marble that passes through the system represents a trial consisting of as many random events as there are rows in the system.

Law of Large Numbers

The Law of Large Numbers says that when a random process, such as dropping marbles through a Galton board, is repeated many times, the frequencies of the observed outcomes get increasingly closer to the theoretical probabilities.

The Kissing Circle

The Osculating Circle, or "Kissing" Circle, is a way to measure the curvature of a line.

Poincaré Disk

The Poincaré Disk is a flat map of hyperbolic space.

Amplitude

The amount of displacement, as measured from the still surface line, is called a wave's amplitude.

The Prime Number Theorem

The answer to the question of why the primes occur where they do on the number line has eluded mathematicians for centuries. Gauss's Prime Number Theorem is perhaps one of the most famous attempts to find the "pattern behind the primes."

Transfinite

The cardinality of sets that cannot be put into one-to-one correspondence with the counting numbers, such as the set of real numbers, is referred to as c. The designations A_0 and c are known as "transfinite" cardinalities.

a divided by b

The expression a/b means

Exponents

The expression a^m means a multiplied by itself m times. The number a is called the base of the exponential expression and the number m is called the exponent. The exponent m tells us to repeat the base a as a factor m times.

each whole number can be uniquely decomposed into products of primes.

The fundamental theorem of arithmetic says that

Hypercube

The hypercube is the four-dimensional analog of the cube, square, and line segment. A hypercube is formed by taking a 3-D cube, pushing a copy of it into the fourth dimension, and connecting it with cubes. Envisioning this object in lower dimensions requires that we distort certain aspects.

Bijection

The identification of a "one-to-one" correspondence--enables us to enumerate a set that may be difficult to count in terms of another set that is more easily counted.

division

The inverse of multiplication

Discrete

The multitude concept presented numbers as collections of discrete units, rather like indivisible atoms.

Expected Value

The notion of expected value, or expectation, codifies the "average behavior" of a random event and is a key concept in the application of probability.

Problem of the Points

The solutions to this gambling dilemma is traditionally held to be the start of modern probability theory.

Standard Deviation

The standard deviation is a way to measure how far away a given individual result is from the average result.

Invarient

The state of appearing unchanged.

Torus

The surface of a standard "donut shape" is known as a torus.

Axiomatic Systems

The system that Euclid used in The Elements—beginning with the most basic assumptions and making only logically allowed steps in order to come up with propositions or theorems—is what is known today as an axiomatic system.

Wave Equation

The wave equation uses second derivatives to relate acceleration in space to acceleration in time.

The Additive Identity Property

The whole number zero is called the additive identity. If a is any whole number, then a + 0 = a.

Galois Theory

This area of mathematics relates symmetry to whether or not an equation has a "simple" solution.

The Riemann Hypothesis

This famous, as yet unproven, result relates to the distribution of prime numbers on the number line.

Fundamental Theorem of Arithmetic

This important result says that every natural number greater than one can be expressed as a product of primes in exactly one way.

Commensurability

This means that for any two magnitudes, one should always be able to find a fundamental unit that fits some whole number of times into each of them (i.e., a unit whose magnitude is a whole number factor of each of the original magnitudes)—an idea known as commensurabilty.

Stereographic Projection

This method can create a flat map from a curved surface while preserving all angles in any features present.

The BML Traffic Model

This model is at the forefront of probability research. Mathematicians use it to model traffic patterns in an attempt to understand flow rates and gridlock, among other things.

Noether's Theorem

This result relates conserved physical quantities, like conservation of energy, to continuous symmetries of spacetime.

Cayley's Theorem

This result says that the symmetries of geometric objects can be expressed as groups of permutations.

Answer the Question

This step is easily overlooked. For example, the problem might ask for Jane's age, but your equation's solution gives the age of Jane's sister Liz. Make sure you answer the original question asked in the problem. Your solution should be written in a sentence with appropriate units.

Normal Distribution

This ubiquitous result describes the outcomes of many trials of events from a wide array of contexts. It says that most results cluster around the average with few results far above or far below average.

Cardinality

Three is the common property of the group of sets containing three members. This idea is called "cardinality," which is a synonym for "size." The set {a,b,c} is a representative set of the cardinal number 3.

1. find a relationship between the first and second numbers. 2. Then we see if the relationship is true for the second and third numbers, the third and fourth, and so on.

To describe and extend a numerical pattern

Genus

Topological objects are categorized by their genus (number of holes). The genus of a surface is a feature of its global topology.

Topology

Topology, originally known as analysis situs—roughly, "geometry of position," seeks to describe what is fundamental about shape in general.

Periodic Function

Trigonometric functions, such as sine and cosine, are useful for modeling sound waves, because they oscillate between values—they are periodic.

Equivalent Equations

Two equations if they have the same solution set.

Grouping Symbols

Use parentheses, brackets, or curly braces to delimit the part of an expression you want evaluated first.

The Set of Whole Numbers

W = {0, 1, 2, 3, 4, 5, . . .} is called

Prime Deserts

We can think of the space between primes as "prime deserts," strings of consecutive numbers, none of which are prime.

Comparison Property

When comparing two whole numbers a and b, only one of three possibilities is true: a < b or a = b or a > b.

One equal sign per line

When writing mathematical statements, follow the mantra:

Frequency

Whether or not we hear waves as sound has everything to do with their frequency, or how many times every second the molecules switch from compression to rarefaction and back to compression again, and their intensity, or how much the air is compressed.

per line

Writing Mathematical equations, arrange your work one equation

Solve the Equation

You must always solve the equation set up in the previous step.

Set up a Variable Dictionary.

You must let your readers know what each variable in your problem represents. This can be accomplished in a number of ways: • Statements such as "Let P represent the perimeter of the rectangle." • Labeling unknown values with variables in a table. • Labeling unknown quantities in a sketch or diagram.

Additive Inverse:

a + (−a) = (−a) + a = 0

Additive Identity:

a + 0 = 0 + a = a

Commutative Property of Addition:

a + b = b + a

Multiplicative Identity:

a · 1 = 1 · a = a

Multiplicative Inverse:

a · 1/a = 1/a · a = 1

Commutative Property of Multiplication:

a · b = b · a

a × b + a × c

a × (b + c)

Division is not Commutative

a ÷ b does not have to be the same as b ÷ a.

Distributive Property:

a(b + c) = a · b + a · c a(b − c) = a · b − a · c

Modular Arithmetic

also known as "clock math," incorporates "wrap around" effects by having some number other than zero play the role of zero in addition, subtraction, multiplication, and division.

Solution

an equation is a numerical value that satisfies the equation. That is, when the variable in the equation is replaced by the solution, a true statement results.

Primes

are the fundamental building blocks of arithmetic.

set

collection of objects. list all the objects in the set and enclosing the list in curly braces.

Multiplying both Sides of an Equation by the Same Quantity

does not change the solution set. That is, if a = b, then multiplying both sides of the equation by c produces the equivalent equation a · c = b · c, provided c = 0.

Dividing both Sides of an Equation by the Same Quantity

does not change the solution set. That is, if a = b, then dividing both sides of the equation by c produces the equivalent equation a/c = b/c , provided c = 0.

A prime number

has no factors other than 1 and itself

A number is divisible by 5

if its final digit is a 0 or 5.

A number is divisible by 10

if its final digit is a 0.

A number is divisible by 3

if the sum of its digits is divisible by 3 (ex: 3591 is divisible by 3 since 3 + 5 + 9 + 1 = 18 is divisible by 3).

A number is divisible by 9

if the sum of its digits is divisible by 9 (ex: 3591 is divisible by 9 since 3 + 5 + 9 + 1 = 18 is divisible by 9).

The index (which becomes the exponent when translating) is the number of times you multiply the number by itself to get radicand.

index p radicand

Hamilton Cycle

is a path that visits every node in a graph and ends where it began.

Variable

is a symbol (usually a letter) that stands for a value that may vary.

perimeter

is the length around an object. Perimeter is used to calculate such things as fencing around a yard, trimming a piece of material, and the amount of baseboard needed for a room. While for a perimeter it is not necessary to have a formula since it is always just calculated by adding the lengths of all sides, sometimes a formula can be useful. If we have a rectangle with width w and length l, the formula for the perimeter is P = 2w + 2l.

De Bruijn Sequence

is the shortest string that contains all possible permutations of a particular length from a given set.

Equation

mathematical statement that equates two mathematical expressions.

≈

means approximately equal.

positive

negative ÷ negative = positive ÷ positive =

negative

negative ÷ positive = positive ÷ negative =

Sign Rules for Division

negative ÷ positive = negative positive ÷ negative = negative negative ÷ negative = positive positive ÷ positive = positive

In Euclidean four-space

our standard notions of Pythagorean distance and angle via the inner product extend quite nicely from three-space.

counting numbers

positive integers are

bar graph

used to display measurements. the measurement was taken is placed on the horizontal axis, and the height of each bar equals the amount during that year.

Multiplication

×, ( )( ), ·,* 1. Multiply the numbers (ignoring the signs) 2. The answer is positive if they have the same signs. 3. The answer is negative if they have different signs. 4. Alternatively, count the amount of negative numbers. If there are an even number of negatives the answer is positive. If there are an odd number of negatives the answer is negative.