Reviewer in Mathematics in the Modern World PDF

Reviewer in Mathematics in the Modern -many natures rhythms are most likely World similar to a heartbeat, while others are like breathing Chapter 1: Mathematics in our Modern World...

Reviewer in Mathematics in the Modern -many natures rhythms are most likely World similar to a heartbeat, while others are like breathing Chapter 1: Mathematics in our Modern World Pattern of Texture Topic: The Nature of Mathematics -exists as a literal surface that we can feel, see, and imagine. Stewart’s Nature Number -can be bristly, and rough, but it can Form also be smooth, and hard. Measure Pattern Geometric Patterns Mathematics is a/an: -consists of a sense of shapes that are typically repeated Study of pattern Language -regularities in the natural world that are Process of thinking repeated in a predictable manner Art -usually visible on cacti and succulents Set of problem-solving tools Pattern Found in Nature Different Kinds of Pattern Waves and Dunes Pattern of Visuals -any form of disturbance that carries -often predictable, never quite energy as it moves repeatable, and often contain fractals -different kinds of mechanical waves -can be seen from seeds and pinecones which propagate through a medium, to the branches, and leaves air, or water, making it oscillate as -visible in self-similar replication of trees, waves pass by ferns, and plants -wind waves, surface waves that create Pattern of Flows the chaotic patterns of the sea -found in water, stone, and even in the -water waves are created by energy growth of trees. passing through water causing it to move in a circular motion. -present in meandering rivers with the repetition of undulating lines Spots and Stripes Pattern of Movement -examples are patterns like spots on the skin of a giraffe, and stripes are visible -pattern in locomotion extends to the on the skin of a zebra scuttling of insects, the flights of birds, the pulsations of jelly fish, and also wave Spirals like movements of fish, worms, and -pattern exist on the scale of the cosmos snakes. to the minuscule forms of microscopic Pattern of Rhythm animals on earth -most basic pattern in nature -also, common and noticeable among plants and some animals -appear in many plants such as 1. Arithmetic Sequence- sequence pinecones, pineapples, and sunflowers of numbers that follows a definite pattern -in animals, ram and kudu also have Example: 2, 4, 6, 8, 10; d=2 spiral patterns on their horns. Formula: an = a1 + (n-1) d Symmetries 2. Geometric Sequence- is a mathematical sequence of non- -if a figure can be folded or divided into zero numbers where each term two with two halves which are the after the first is found by same, such figure is called symmetrical multiplying the previous one by a figure fixed number called the common -used to classify and organize ratio. information about patterns by classifying Example: 2, 8, 32, 128; r=4 the motion or deformation of both Formula: an = a1 rn-1 pattern structures and processes. 3. Harmonic sequence- the reciprocal of arithmetic Kinds of Symmetries sequence and comes with a 1. Reflection- line symmetry or mirror fraction symmetry, captures symmetries Example: ½, ¼, 1/6, 1/8, 1/10,... when the left half of a pattern is Formula: an = 1/ [a1 + (n-1) d] the same as the right half. 4. Fibonacci Sequence- named 2. Rotations- captures symmetry after an Italian mathematician when it still looks the same after Leonardo Pisano Bigollo (1170- some rotation (of less than one 1250); discovered the sequence full turn); degree of rotational while studying rabbits. The symmetry of an object is sequence is organized in a way a recognized by the number of number can be obtained by distinct orientations in which it adding the previous number looks the same for each rotation Example: 1, 1, 2, 3, 5, 8, 13,... 3. Translations- exist in patterns that Chapter 2: Characteristics and we see in nature and in man- Conventions in the Mathematical made objects; acquire Language symmetries when units are repeated and turn out having Topic: Characteristics of Mathematical identical figures, like the bees' Language honeycomb with hexagonal tiles. 1. Precise Symmetries in Nature includes: 2. Concise 3. Powerful 1. Human body 2. Animal movement 3. Sunflower 4. Snowflakes 5. Honeycomb/beehive Importance of Mathematical Language 6. Starfish Major contributor to overall The Fibonacci Sequence comprehension  Vital for the development of 4. Infinite Set- element in a given set Mathematics proficiency is not countable Enables both the teacher and 5. Cardinal number- number of students to communicate elements in a given set mathematical knowledge with 6. Equal Set- equal number of precision cardinalities and the elements are identical Comparison of Natural Language into 7. Equivalent Set- have the exact Mathematical Language number of elements Nouns in Mathematics could be 8. Universal Set- set of elements fixed things such as numbers or under discussion expressions with numbers. 9. Joint Set- have common Verbs could be equal sign or elements inequalities 10. Disjoint Set- have no common Pronouns could be variable elements Expressions and Sentences Two Ways of Describing a Set A mathematical sentence expresses a 1. Roster or Tabular method- done complete mathematical thought about by listing or tabulating the relation of mathematical object to elements of the set another mathematical object. 2. Set Builder Notation Four Basic Concepts Subsets- means that every element of A is also an element of B. Sets and Subsets Note: The number of subsets of a given Georg Cantor use the word set as a set is given by 2n , where n is the number formal mathematical term was of elements. introduced in 1879. Operations on Sets A set is a collection of well-defined objects. 1. Union sets 2. Intersection of sets Examples: 3. Difference of sets 1. A set of counting numbers from 1- 4. Compliment on sets 10 5. Cartesian product A= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Venn Diagram- illustration of relationship 2. A set of English alphabets from a- between among sets, groups of objects e that share something in common. A= {a, b, c, d, e} Terminologies of Sets 1. Unit Set- a set contains of only one element Functions and Relations 2. Empty set or Null set- a set that Relation- set from set X to Y is the set of has no element ordered pairs of real numbers (x, y) such 3. Finite Set- elements in a given set that to each element x of the set x there is countable corresponds at least one element of the Inductive Reasoning- type of reasoning set y that forms a conclusion based on examination of specific examples. Function- A function is a relation in which every input is paired with exactly 1. Observe and look for pattern one output. (x, y); where x is the domain 2. Analyze what is really happening and y is the range in the pattern 3. Make a conjecture -No repeating value of x Examples One-to-one (function) 1. Observe and look for pattern: One-to-many (not a function) 2, 4, 12, 48, 240 Many-to-one (function) 2. Analyze what is really happening in the pattern: Binary Operations The numbers are multiplied by 2, Let g be a non-empty set. An operation then 3, then 4, and then 5. * is said to be binary operation on G if 3. Make a conjecture: for every element, a, b is in G that is a, b Therefore, the answer is 1,140. ∈ G, the product a * b ∈ G. Deductive Reasoning- process of Closed- a set is closed under operation reaching specific conclusion by if the operation assigns to every ordered applying general ideas, or assumptions, pair of elements from the set an procedure or principle or it is a process element of the set. of reasoning logically from given statement to a conclusion. Properties of Binary Operations 1. General ideas 1. Associative 2. First premise that fits within 2. Commutative general truth 3. Identity 3. Second premise that fits first 4. Inverse premise Cayley Table- a square grid with one 4. Specific conclusion row and one column for each element Example: in the set. The grid is filled in so that the element in the row belonging to x and First premise: All positive counting the column belonging to y is x * y. numbers whose unit digit is divisible by two are even numbers. Second premise: A positive counting number 1,236 has a unit digit of 6 which is divisible by two. Chapter 3: Problem-Solving and Conclusion: therefore, 1, 236 is an even Reasoning number. Topic: Inductive and Deductive Intuition, Proof, and Certainty Reasoning Intuition- it is an immediate understanding or knowing something. Proof and certainty- a proof is an 6. Therefore, If x is a number with 5x + inferential argument for a mathematical 3 = 33, then x = 6. QED statement while proofs are an example of mathematical logical certainty. Kinds of Proof A mathematical proof is a list of Direct Proof- a mathematical statements in which every statement is argument that uses rules of inference one of the following: to derive the conclusion from the (1) an axiom premises. (2) derived from previous statements by Example: If a and b are both odd a rule of inference integers, then the sum of a and b is an even integer. (3) a previously derived theorem 1. Assume that a and b are both Theorem: A statement that has been integers. By definition of odd proven to be true. integers, we can say that a = 2m + 1 Proposition: A less important but and b = 2n + 1, such that m and n nonetheless interesting true statement. are elements of integer. Lemma: A true statement used in 2. Substitute the values of a and b to proving other true statements (that is, a a + b, which will result to a + b = 2m less important theorem that is helpful in + 1 + 2n +1 = 2m + 2n + 2. the proof of other results). Corollary: A true statement that is a 3. Factor out 2 from 2m + 2n + 2, simple deduction from a theorem or resulting to a +b = 2 (m +n + 1). proposition. 4. Represent m + n + 1 with another Proof: The explanation of why a variable k, such that k is an element statement is true. of an integer, resulting to a + b = 2k. Presenting a Proof: 5. By definition of even integers, the sum of a and b is an even integer. Example 1: Prove (in outline form) QED. that “If x is a number with 5x + 3 = 33, then x= 6” 1. Assume that x is a number with 5x + 3 = 33. Indirect proof- a statement to be 2. By APE, 5x + 3 – 3 = 33 – 3 proved is assumed false and if the assumption leads to an impossibility, 3. Then it will result to 5x = 30. then the statement assumed false 4. By MPE, 1/5 (5x = 30). has been proved to be true. 5. So x = 6. Example: Prove that, if x is divisible by 6, then x is divisible by 3. 1. Assume that x is not divisible by 3. By definition of numbers divisible by 3, we can say that x = 3k, such k is an element of an integer. 2. Then, we can say that if x = 3k, then x = 3 (2k). 3. Simplifying, we’ll have x = 6k. 4. Based on the definition of numbers divisible by 6, then we can say that x is not divisible by 6. 5. Hence, if x is divisible by 6, then x is divisible by 3.