Mathworld - lecture notes PDF

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Math World Mathematics – is science of numbers, problem solving, theorem solving, science of measuring, science of patterns. -

Format system for recognizing classifying and expecting patterns. To organize and systemize about patterns. The simplest mathematical objects are numbers. The simplest of nature patterns are numerical.

What Is mathematics for? -

Understand patterns Organize underlying patterns To protect the natures behavior To control nature To make practical use of what we learned about our world.

Complex number systems a. Natural Numbers – Positive integers (whole numbers) Symbol: N Example: (1, 2, 3, 4 . . . . ) b. Whole Numbers – Numbers without fractions Example: (0, 1, 2, 3, 4, . . . ) c. Integers – a whole number that can be positive, negative or zero. Example: (…. -2 ,-1, 0, 1, 2 …) d. Rational Numbers – Fractions, terminating and repeating decimals. e. Irrational Numbers – Non – terminating and non – repeating decimals; pie f. Real Numbers – Rational, Integers, Whole and Natural g. Imaginary Numbers – i, 2i, -3-7i, etc (i = π ) Set of Numbers

Levels of Measurements a. Nominal – Variable with no inherent order can be ( I, ii, iii) is enumeration category. b. Ordinal – Rank or Order ex. Mild, Moderate. Can be compared for equality but not how much greater or less. c. Interval – Variable are ordered as in ordinal, differences between values. Calendar dates and temperatures, Celsius, addition&subtraction, cognitive skills, test score.. d. Ratio – Variables with all properties (zero point, kelvin, age, +, -, x, ÷ (possible zero)

The Nature of Mathematics Fibonnaci – he discovered the pattern of the sequence of numbers from the set (1,1,2,3,5,8,13….) Leonardo Pisano (Real Name of Fibonacci in Italian) -

Means Leonardo of Pisa, because he was born in Pisa, Italy around 1175.

Fibonacci – shortened word for the Latin term “filius Bonacci” which stands for “son of Bonaccio” Guglielmo Bonaccio – father of Leonardo

Johannes Kepler – German Mathematician and Astronomer -

Observed that dividing a Fibonacci number by a number immediately before it in the ordered sequence yields a quotient approximately equal to 1.6180339887… or approximately 1.618 this is called Golden Ratio and denoted by a symbol φ (phi).

Divine Proposition or Golden Mean – other name for Golden Ratio Golden Rectangle – A rectangle can be drawn of such a shape that if it is cut into square and a rectangle, the smaller rectangle will be similar in a shape to the larger rectangle. Iteration – repeated application of an operation on a given function over and over again.

*Mathematical Language can describe a subset of the real world using only the symbols above. Left Brain Hemisphere – responsible for controlling language and part of the brain in change of tasks involving mathematics. Mathematical Sentence – must state a complete thought. Mathematical Expression – is a name given to a mathematical object of interest. Truth of Sentence- mathematical sentence may either be true or false but not both. Example 1. Write as English sentence and say whether they are true or false. a . ∀x ∈ R , x 2 ≥ 0 -

For any real number x, its square is greater than or equal to 0. TRUE

Mathematical Language and Symbols

b. ∀x , y ∈ R , ( x+ y ) 2=x 2+2 xy + y 2

Mathematical Language – precise w/c means it is able to make very fine distinctions or definitions among a set of mathematical symbols.

- For any real numbers x and y, the square of their sum is equal to the sum of their squares plus twice their product. TRUE

∑ - The sum of ⱻ - There exists ꓯ - For every (for any)

∈ - element of (or member of) - not an element of ⊆ - subset of ⇒ - if…, then ⇔ - if and only if R – set of real numbers N – set of natural numbers Z – set of integers Q – set of rational numbers

∞ - infinity

m , n ∈ Z , m −n ≤ m + n c. ∃ - There exist integers m and n such that m minus n is less than or equal to m plus n. TRUE d. ∀a , b ∈ Q , ab =0 ⇒a=0 b=0 - For any rational numbers a and b, if their product is zero then either a or b equals O. TRUE 2

e. +¿ , ∃y ∈ R , ¿y = x ∀x ∈ Z - For every positive integer x, there exists a real number y, such that y 2=x . TRUE Example 2. Write as mathematical sentence. a. Ten is the square root of 100.

√ 100=10 b. Ten is greater than 9. 10 ¿ 9

c. Ten is an even number. 10 ∈(2 n , n∈ N ) d. Ten is a multiple of 5. 10 ∈ ( 5 n , n ∈ N ) Is – could mean equality, inequality, or membership in a set. Type of Numbers a. Cardinal Numbers – are used for counting and answer the question “how many?”. b. Ordinal Numbers – tell the position of a thing in a term of first, second, third, etc. c. Nominal Numbers – are used only as a name or to identify something. Unary Operation – involve only one value. Binary Operation – take two values, and include the operations of addition, subtraction, multiplication, division, and exponentiation.

performed on a group of two numbers added or subtracted together. Ex: zx ± zy e. Identity Elements of Binary Operations – is the number that you add to any real number and the result will be the same real number. The only number that satisfies this property is the number 0 for addition. Ex: 5+0=0+5=5 f. Inverse of Binary Operations Ex: x + (-x) = -x + x = 0 1 1 x = x =1 x x

()()

Logic – Allows us to determine the validity of arguments in and out of mathematics. -

Properties of Binary Operations a. Closure of Binary Operations – the product and the sum of any two real numbers is also a real number. ∀x , y ∈ R , x + y ∈ R Ex: and ( x)( y ) ∈ R b. Commutativity of Binary Operations – addition and multiplication of any two real numbers is commutative. Ex: x + y = y + x (addition) (x)(y) = (y)(x) (multiplication) c. Associativity 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. Ex: (1 + 2) + 3 = 1 + (2 + 3) (2)(3)4 = 2(3)(4) d. Distributivity of Binary Operations – Distributivity applies when multiplication is

Illustrate the importance of precision and conciseness of the language of math.

Propositions – is a statement which is either True of False. -

Must express a complete thought. A declarative.

Example: a. 9 is a prime number. - False. Prime numbers have no other factors than 1 itself. 9 can be expressed as 3(3). b. 5 + 3 = 8 - TRUE 2 2 c. x +y ≥0 - TRUE d. 10...