MMW Midterms

Mathematics

Explains a phenomenon, behavior, or structure of nature through patterns, constants which are represented by numbers and equations; prediction.

Fibonacci Numbers

Numbers are characterized by every number after the first two is the sum of the two preceding ones and also known as nature’s numbering system. That is… f(n) = f(n -1) + f(n - 2)

1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next)

Leonardo Pisano (Leonardo of Pisa)

• Born in Pisa, 1175

• Wrote the “Liber Abaci,” introducing Arabic numerals into European Math

• Made the Fibonacci Sequence

• He observed numbers in nature; the most popular is the Fibonacci numbers.

Fibonacci

Short for the Latin of "filius Bonacci" which means "son of Bonacci"

Fibonacci Spiral

Series of connected quarter circles drawn inside an array of squares with Fibonacci Numbers for dimensions. It gets closer and closer to a Golden Spiral as it increases in size because of the ratio (values from the Golden Ratio).

Golden Ratio

The ratio of two consecutive Fibonacci numbers as n becomes large. Where L is 1.6180339887...

Golden Rectangle

Ratio of length to width

Language

A systematic means of communicating ideas or feelings by the use of conventional symbols, sounds, or marks having understood meaning. Facilitates communication and clarifies meaning. Allows people to express themselves and maintain their identity

Mathematics is also a language due to the use of symbols (symbol system).

• Can describe a subset of the real world using only these symbols. e.g., In Physics - Free falling bodies, speed, and acceleration; In Biology - modeling diseases.

  • It describes abstract structures. e.g., Pure Mathematics - Abstract Algebra, Linear Algebra, Real Analysis, and Complex Analysis.

Precise

Characteristics of Math Language that is able to make very fine distinctions

Concise

Characteristics of Math Language that is able to say things briefly

Powerful

Characteristics of Math Language that is able to express complex thoughts with relative ease

English Language

Contains a complete thought. Has a subject that is a noun or a clause, and a predicate.

Mathematical Language

A sentence must state a complete thought

Mathematical Expression

Mathematical object of interest, it is a group of number/s with or without mathematical operations including a correct arrangement of mathematical symbols used to represent a mathematical object.

Truth of Sentences

A Mathematical sentence must state a complete thought. And it may be either true, false, and sometimes true / sometimes false. But never both.

Cardinal Numbers

Used for counting. “How many?”

Ordinal Numbers

Tells the position of a thing in the list. “First, second, third, ...”

Nominal Numbers

Used only as a name for identification. “Zip Code of Bacolod City: 6100”

Unary Operations

Accepts only one value or operand. When “plus” and “minus” signs are attached before a single number and do not mean addition or subtraction

Binary Operations

It takes two real numbers as arguments to produce another real number. If the + and - signs act on two operands, then it is called a binary operation. 

Closure of Binary Operations

Properties of Binary Operations: The product and the sum of any two real numbers is also a real number.

Commutativity of Binary Operations

Properties of Binary Operations: Addition and multiplication of any two real numbers is commutative, that is in mathematical symbols these are written,

Associativity of Binary Operations

Properties of Binary Operations: Given any three real numbers you may take any two and perform addition or multiplication as the case maybe and you will end with the same answer.

Distributivity of Binary Operations

Properties of Binary Operations: Applies when multiplication is performed on a group of two numbers added or subtracted together

Identity Elements of Binary Operations

Properties of Binary Operations: An element of the set of real numbers is an identity element for addition

Inverses of Binary Operations

Properties of Binary Operations: For addition, you add the negative of a number to get the identity element. For multiplication, we multiply with its reciprocal.