Mathematics In The Modern World Midterms-Finals

Qualitative Variables

- a variable that yield observations by which individuals can be categorized according to some characteristic or quality.

Quantitative Variables

- a variables that yield observations which can be measured.

Ordinal Level

- classifies data into categories that can be ranked, however, precise differences between ranks do not exists. Examples: educational attainment (elementary, high school, college), socio-economic class (elite, middle, low), rating scale (excellent, very good, satisfactory, fair, poor), judging (1st, 2nd, 3rd , etc.)

positive relationship

- exists when both variables increase and decrease at the same time.

Negative relationship

- exists when one variable increases while the other variable decreases.

Ratio Level

- has all the characteristics of the interval level of measurement and the zero point is meaningful (which represents the complete absence of characteristics being measured), and the ratio between two numbers is meaningful. Examples: height, weight, income, distances, time, age, area

parallel lines

- lines that do not intersect or touch each other at any point.

Multiple relationship

- many variables are under study

Discrete Variables

- quantitative variables whose values are obtained from counting.

Continuous Variables

- quantitative variables whose values are obtained from measurements. It can assume all values between any two specific values.

Interval Level

- ranks data and precise differences between units of measure do exists but there is no meaningful or true zero. It includes the ranking characteristics of the ordinal level of measurement and specifies that the distance between numbers is the same. Zero is arbitrary Examples: temperature, IQ test scores, NSAT scores

Nominal Level

- refer to data that can only be classified into categories. No ranking or order can placed on the data Examples: color (black, blue, yellow), gender (male, female), religion (Catholic, Protestant, etc.), field of specialization (English, Mathematics, Psychology, etc.), marital status (single, married, widowed, separated)

measurement

- the assignment of numbers to properties of objects, events or persons according to logically accepted rule.

Dependent Variable

- the criterion variable, the response or the effect, a value depends on some other variables called the independent variables.

Independent Variable

- the predictor variable or the cause which can be manipulated to determine its effect on the dependent variable.

Simple relationship

- two variables are under study

Eccentricity of graph

-It is defined as the maximum distance of one vertex from other vertex.The maximum distance between a vertex to all other vertices is considered as the eccentricity of the vertex. It is denoted by e(V).

Measures of Central Tendency: Median

-the middle or central value in a data set when arranged in the order of magnitude. -divides the data set into two equal parts. -can be used when the data is ordinal. -can be used to determine whether the data values fall into the upper (lower) half of the distribution can be used to find average of an open-ended distribution.

Measures of Central Tendency: Mode

-the most typical value which occurs most frequently in a data set. -can be used when the data is nominal. -not always unique and does not always exist for a data set. -value (class) with the largest frequency.

Cryptology

= cryptography + cryptanalysis

Cryptanalysis

= the science (art) of breaking encryption

Cryptography

= the science (art) of encryption

Logic

=Commonly known as the science of reasoning. =Definition: Methods of reasoning, provides rules and techniques to determine whether an argument is valid

Mathematics Behind Graphics

A lot of game engines make use of physics as well to compute things like light scattering in 3D games. I can tell you from experience that these are really complicated and ugly integrals, and most computers use numerical algorithms (like finite difference methods) to solve them.

circuit.

A path that starts and ends at the same vertex is called a

logically implies

A statement formula A is said to ______ a statement formula B if the statement formula A → B is a tautology.

logically equivalent

A statement formula A is said to be ______ to a statement formula B if the statement formula A ↔ B is a tautology.

contradiction

A statement formula A is said to be a _____ if the truth value of A is F for any assignment of the truth values T and F to the statement variables occurring in A

Tautology

A statement formula A is said to be a _______ if the truth value of A is T for any assignment of the truth values T and F to the statement variables occurring in A.

Caesar Cipher

ABCDEFGHIJKLMNOPQRSTUVWXYZ Key = 3 DEFGHIJKLMNOPQRSTUVWXYZABC Example Plaintext: COLLEGE Encryption: Shift by KEY = 3 Ciphertext: FROOHJH Decryption: Shift backwards by KEY = 3

Computer Science

Data compression schemes are used to compress files by exploiting patterns in the data. For instance, instead of writing 'aaaaa', a computer could simply mark that region in memory as 5a, indicating that this section has five a's, cutting down the number of characters needed to express that from 5 to just 2.

Mathematics in Commerce

Discount • Banking • Foreign Exchange • Stock and Share • Arithmetic • Profit & Loss, Percentage, Ratio and Proportion, Time problem

Graph Theory

Graphs are nothing but connected nodes(vertex). So any network related, routing, finding relation, path etc. related real life applications use graphs.

Number Theory

It is the branch of pure Math concerned with properties of numbers. • It is used for creating codes for ATM/ Credit Cards. • Password and credit card numbers are protected using number theory (cryptography)

Math in Engineering

Mathematics is involved in everything an engineer does, whether it is working out how much concrete is needed to build a bridge,or determining the amount of energy to power a car or power to move the LRT

cooking

Measuring (3/4 cups, 1 teaspoon, 1 tablespoon) • Conversion (kilogram o gram, liter to ml, centigrade to fahrenheit)

Truth value

One of the values "truth" or "falsity" assigned to a statement True is abbreviated to T or 1 False is abbreviated to F or 0

Mathematics Behind Google

PageRank Algorithm- used graph theory , linear algebra and probability PageRank score of a webpage represents the probability that a random Web surfer chooses to view the webpage.

Polya's Four-step in Problem Solving

Step 1 understand the problem step 2 devise a plan step 3 carry out the plan step 4 look back

Stock and Share

Stocks are sold by Brokers. The Cost Price = Market value + Brokerage

negation

Symbol ~ is called "not" ~p is read as as "not p"

disjunction

The ________ of p and q, written p ∨ q , is the statement formed by joining statements p and q using the word "or".

conjunction

The _______of p and q, written p ^ q , is the statement formed by joining statements p and q using the word "and"

Additive Inverse in mod

The additive inverse of 𝑎 in 𝑚𝑜𝑑 𝑏 is the smallest whole number 𝑑 such that (𝑎+𝑑)(𝑚𝑜𝑑 𝑏)=0. Examples: In 𝑚𝑜𝑑 4, the additive inverse of 3 is 1 since (3+1)=4 and 4(𝑚𝑜𝑑 4)=0. In 𝑚𝑜𝑑 7 , the additive inverse of 11 is 3 since (11+3)=14 and 14(𝑚𝑜𝑑 7)=0.

Multiplicative inverse in mod

The multiplicative inverse of 𝑎 in 𝑚𝑜𝑑 𝑏 is the smallest whole number 𝑑 such that (𝑎𝑑)(𝑚𝑜𝑑 𝑏)=1. Examples: In 𝑚𝑜𝑑 4, the multiplicative inverse of 3 is itself since (3∙3)=9 and 9(𝑚𝑜𝑑 4)=1. In 𝑚𝑜𝑑 7 , the multiplicative inverse of 11 is 2 since (11∙2)=22 and 22(𝑚𝑜𝑑 7)=1.

implication or condition.

The statement "if p then q" is called _________ The implication "if p then q" is written p → q

biimplication or biconditional

The statement "p if and only if q" is called the _________ of p and q The biconditional "p if and only if q" is written p ↔ q

Foreign Exchange

What are the rates of each currency when compared to Peso?

Modular Arithmetic

When we divide two whole numbers we will have an equation that looks like the following: 𝑎/𝑏=𝑞 remainder 𝑟 (𝑏≠0). Sometimes, we are only interested in what the remainder is when we divide 𝑎 by 𝑏. .For these cases there is an operator called the modulo operator (abbreviated as mod). Using the same 𝑎,𝑏,𝑞,𝑟 as above, we would have: 𝑎 (mod 𝑏)=𝑟. We would say this as 𝑎 modulo 𝑏 is equal to 𝑟. Where 𝑏 is referred to as the modulus.

Telling Time

You need to know that there are 24 hours in a day, that we split those days into two equal-sized 12-hour halves, that each hour is 60 minutes, and that each minute is 60 seconds.

Quantification

______ expresses the extent to which a predicate is true over a range of elements The words all, some, many, none, and few are used in quantification

expression

`nouns' of mathematics (used to name mathematical objects of interest)

bilateral symmetry

a symmetry in which an object has two sides that are mirror images of each other.

radial symmetry

a symmetry where there is a center point and numerous lines of symmetry could be drawn.

fractals

are objects in which the same patterns occur again and again at different scales and sizes

Adjacent Edges

are two edges that intersect at a vertex.

Adjacent Vertices

are two vertices that are joined by an edge.

Descriptive measures

are values that are used to describe and summarize sets of data. These include:

compound propositions or propositional functions

can be obtained from old ones by using symbolic connectives

Linear Cipher

combines the Multiplication and the Caesar Cipher. To encode, we first multiply the value of the plain letter P by a and secondly add b. Thus, the encryption function is  Y = (A X + B) mod 26. To decode we have to first undo the addition performed in the encryption process. Afterwards, we undo the multiplication.

truth table

displays the relationships between the truth values of propositions.

Shapes

geometry is the branch of mathematics that describes _______

Fibonacci spiral

if you construct a series of squares with lengths equal to fibonacci numbers and trace a line through the diagonals of each square

Euler Circuit

is a circuit that travels through every edge of a graph.

Statement or proposition

is a declarative sentence that is either true or false, but not both Lowercase letters denote propositions

Scatter plot (scatter point diagram)

is a graph of independent and dependent variables in correlation or regression analysis.

Intuition

is a mathematical belief without being formalized and proven directly and serves as an essential part of mathematics.

Correlation coefficient

is a measure used to determine whether two or more variables are related and also to determine the strength of the relationship between or among the variables

Linear correlation

is a method used to describe the relationship between two quantitative (scale) variable which measures the strength of the association of the variables under study.

Euler Path

is a path that travels through every edge of the graph (once and only once).

path

is a sequence of vertices such that each vertex is adjacent to the next. In a path, each edge can be traveled only once.

Correlation

is a statistical method used to determine whether a relationship exists between variables.

Vigenere Cipher

is an improvement of the Caesar Cipher. Example: Using the keyword "cat" produce shifts of length 2 (=c) for the 1st , 4th , 7th, 10th , ... plain letter, shifts of length 0 (=a) for the 2nd , 5th , 8th , 11th , ... plain letter and shifts of length 19 (=t) for the 3rd , 6th , 9th , 12th , ... plain letter. theappleisinthecornerandthepearistheretoo (plain) catcatcatcatcatcatcatcatcatcatcatcatcatca (key)vhxcpinebuigvhxeokpekcnwvhxrettilvhxtemqo (cipher)

Proof

is an inferential argument for a mathematical statement.

Certainty

is something that is completely certain to happen

The degree of a vertex

is the number of edges at that vertex.

Symmetry

is when a figure has two sides that are mirror images of one another.

Calculus

it helps us to find the area of figures • to know the acceleration of a car moving in a highway with a uniform speed • use to know the movement of an artificial satellite, ship etc. • se to calculate the economical dimension of our basic needs (can goods, noodles, softdrinks)

Standard Deviation

most widely reported measures of dispersion.

Deductive reasoning

reasoning goes from general to specific

Inductive reasoning

reasoning goes from specific to general

Encryption:

scrambling a message or data using a specialized cryptographic algorithm.

sentence

state complete mathematical thoughts

Universal quantification

tells that a predicate is true for every element under consideration

Existential quantification

tells that there is one or more element under consideration from which the predicate is true.

Cipher

the algorithm that does the encryption.

Mean Deviation, MD

the average deviation of all observations from the mean, found by taking the sum of all the absolute values of the deviations divided by the number of observations.

Ciphertext:

the encrypted (scrambled) version of the message.

Plaintext

the message or data before it gets encrypted.

Decryption:

the process of converting ciphertext back to the original plaintext.

Range, R

the set of data is the difference between the highest and the lowest values on the data set. easy to compute but less reliable measure of variability, since influenced by extremely large or small values simply subtract the lowest value (LV) from the highest value (HV) on the data set

Precedence of logical connectives

~ highest ∧ second highest ∨ third highest → fourth highest ↔ fifth highest

Statistics

• Collection • Analysis • Interpretation or Explanation • Presentation of Data • Prediction • Forecasting

Communication Technology

• In wireless communication, information is sent through what is called a channel. • The channel is subject to noise, so that there will be some loss of information • This was analyze using coding theory (with the Galois Theory as the foundation)

Mathematics in Forensic

• mathematics lies behind expert conclusions on a hundred forensic matters from fingerprints to DNA • Measurement, proportion, trigonometry and probability are some of them.

CHARACTERISTICS OF MATHEMATICS LANGUAGE

• precise (able to make very fine distinctions); • concise (able to say things briefly); • powerful (able to express complex thoughts with relative ease).