Math 1F Reviewer PDF

MATH 1F REVIEWER (TERMS ONLY) **Mathematics** is the study of the relationships among numbers, quantities and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics and calculus. **Gary Smith** adopts eight patterns in his landscapes work, namely: scattered, fractured, mosaic,...

MATH 1F REVIEWER (TERMS ONLY) **Mathematics** is the study of the relationships among numbers, quantities and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics and calculus. **Gary Smith** adopts eight patterns in his landscapes work, namely: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial and dendritic. These patterns occur in plants, animals, rock formations, river flow, stars or in human creations (Goral, 2017). **TYPES OF PATTERNS** 1\. **SYMMETRY** - a sense of harmonious and beautiful proportion of balance or an object is invariant to any various transformations(reflection, rotation or scaling.) a.) **Bilateral Symmetry ** : a symmetry in which the left and right sides of the organism can be divided into approximately mirror image of each other along the midline. Symmetry exists in living things such as in insects, animals, plants, flowers and others. Animals have mainly bilateral or vertical symmetry, even leaves of plants and some flowers such as orchids. b.) **Radial --** A five-fold symmetry is found in the echinoderms, the group in which includes starfish (dihedral-D5 symmetry), sea urchins and sea lilies**. CIRCLE** A pattern that is arranged around a central point is called a **radiating pattern. CURVED** 2\. **FRACTALS** - is a never-ending pattern found in nature. Same shape but different in size 3\. **SPIRALS -** A spiral is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Examples of spirals are pinecones. 4.. **Dendritic Pattern -** resembles the branches of a tree and develops gently into a sloping basin with the uniform rock type 5\. **Scattered Pattern -** A scattered pattern is a dispersed settlement pattern which means the buildings are not squished together. 6\. **Mosaic Pattern --** A piece of art or image made from the assembling of small pieces of colarred materials. 7\. **Stripe --** is a line or band that differs in color or tone 8\. **Spots --** Usually consists of dots 9\. **Tessellations --** The tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps In the 19th century, Belgian physicist **Joseph Plateau** examined soap films, leading him to formulate the concept of a minimal surface. German biologist and artist **Ernst Haeckel** painted hundreds of marine organisms to emphasize their symmetry. Scottish biologist **D'Arcy Thompson** pioneered the study of growth patterns in both plants and animals, showing the simple equations could explain spiral growth. In the 20th century, British mathematician **Alan Turing** predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. **Hungarian biologist Aristid Lindenmayer and French American mathematician Benoit Mandelbrot** showed how the mathematics of fractals could create plant growth patterns (Patterns in Nature, 2017). **Leonardo Pisano Bogollo** lived between 1170 and 1250 in Italy. His nickname, **"Fibonacci" roughly means "Son of Bonacci".** Aside from being famous for the Fibonacci Sequence, he also helped spread **Hindu Arabic** numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) through Europe in place of Roman numerals (I, II, III, IV, V, etc). **November 23 --** Fibonacci day **1.1618 --** Golden Ratio The art of applying mathematics to complex real-world problems is called **engineering mathematics**. It combines mathematical theory, practical engineering and scientific computing to address the fast-changing technology. Engineering mathematics can be found in an extraordinarily wide range of careers, from designing next generation high-end cars to inventing robotics and automatic devices. **Language** is the system of words, signs and symbols which people use to express ideas, thoughts and feelings. **Mathematical language** is the system used to communicate mathematical ideas. Numbers, measurements, shapes, spaces, functions, patterns, data and arrangements are regarded as **mathematical nouns,** or objects while **mathematical verbs** may be considered as the four main actions attributed to problem-solving and reasoning. **According to Kenney, Hancewicz, Heuer, Metsisto and Tuttle (2005), these four actions are:** ▪ **Modeling and Formulating:** creating appropriate representations and relationships to mathematize the original problem. ▪ **Transforming and Manipulating:** changing the mathematical form in which a problem is originally expressed to equivalent forms that represent solutions. **▪ Inferring:** applying derived results to the original problem situation, and interpreting and generalizing the results in that light. **▪ Communicating**: reporting what has been learned about a problem to a specified audience. The pictorial representation of relationship and operations of sets is the so-called **Venn-Euler Diagrams or simply Venn Diagrams**. The universal set is usually represented by a rectangle while circles within the rectangle usually represent its subsets. The shaded region in the given diagrams illustrates the sets relation or operation. Mathematics is about ideas -- relationships, quantities, processes, measurements, reasoning and so on. The use of language in mathematics differs from the language of ordinary speech in three important ways, according to Jamison (2000). **▪ First, mathematical language is non-temporal. There is no past, present or future in mathematics.** **▪ Second, mathematical language is devoid of emotional content.** **▪ Third, mathematical language is precise** **Mathematical expressions** consist of terms**.** The term of a mathematical expression is separated from other terms with either plus or minus signs. A single term may contain an expression in parentheses or other grouping symbols. In algebra, **variable or letters** are used to represent numbers**.** The **variable,** also called **literal coefficient**, represents the unknown and makes use of letters. The number with the variable is the **numerical coefficient.** Any single number is called a **constant.** For example, 21(x -- 8) has **1 term,** thus, it is called a **monomial.** On the other hand, 5x + 12y has **two terms,** this Mathematical expression is called **Binomial** while 3x + 2(x + y) -- 36 has **three terms, ,** this Mathematical expression is called **Trinomial** A mathematical expression with **more than two terms** is called **polynomial.** A **trinomial** is a **polynomial.** A **mathematical sentence** combines two mathematical expressions using a comparison operator. These expressions either use numbers, variables, or both. The comparison operators **include equal, not equal, greater than, greater than or equal to, less than, and less than or equal to.** A mathematical expression containing the equal sign is an **equation.** A mathematical expression containing the inequality sign is an **inequality.** An **open sentence** in math means that it uses variables, meaning that it is not known whether the mathematical sentence is true or false. A **closed sentence**, on the other hand, is a mathematical sentence that is known to be either true or false. **Context** refers to the particular topics being studied, and it is important to understand the context to understand mathematical symbols. **Convention** is a technique used by mathematicians, engineers, scientists in which each particular symbol has particular meaning. A **set** is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's **elements or members**) can be numbers, people, letters of the alphabet, other sets, etc. Sets are conventionally named with **capital letters**. There is a simple notation of sets**. Braces** are usually used to specify that the objects written between them belong to a set. The **elements/members** inside the braces are separated by a **comma** There are two ways to describe a set, namely: ▪ In the **Roster/Tabular Method**, the elements in the given set are listed or enumerated, separated by a comma, inside a pair of braces. ▪ In the **Rule/Descriptive Method**, the common characteristics of the elements are defined. This method uses set builder notation where x is used to represent any element of the given set. The following are kinds of sets: ▪ **Empty/Null/Void set** has no element and is denoted by ∅ or by a pair of braces with no element inside, i.e. { }. ▪ **Finite set** has countable number of elements. **▪ Infinite set** has uncountable number of elements. ▪ **Universal set** is the totality of all the elements of the sets under consideration, denoted by U. Two or more set may be related to each other as described by the following: ▪ **Equal sets** have the same elements. **▪ Equivalent sets** have the same number (cardinality) of elements. ▪ **Joint sets** have at least one common element. ▪ **Disjoint sets** have no common element. It can be noted that equal sets are equivalent sets, however, not all equivalent sets are equal sets. A **subset** is a set every element of which can be found on a bigger set. The symbol ⊂ means "a subset of" while ⊄ means "not a subset of". If the first set equals the second set, then it is an **improper subset**. The symbol ⊆ is used to mean an improper subset**.** A **null set** is always a subset of any given set and is considered an improper subset of the given set. Other than the set itself and the null set, all are considered **proper subsets**. The set containing all the subsets of the given set with n number of elements is called the **power set** with 2n number of elements. There are four operations **(Addition, Subtraction, Multiplication, and Division)** performed on sets. Suppose we named the two sets as Set A and Set B, then: ▪ **Union of Sets A and B** (denoted by A ∪ B) is a set whose elements are found in A or B or in both. In symbol: A ∪ B = {x \| x ∈ A or x ∈ B} ▪ **Intersection of sets** A and B (denoted by A ∩ B) is a set whose elements are common to both sets. In symbol: A ∩ B = {x \| x ∈ A and x ∈ B} ▪ **Difference of sets** A and B (denoted by A -- B) is a set whose elements are found in Set A but not in Set B. In symbol: {x \| x ∈ A and x ∉ B} ▪ Complement (**Complements of Set**) of Set A (denoted by A') is a set whose elements are found in the universal set but not in Set A. In symbol: A' = {x \| x ∈ U and x ∉ A ▪In mathematics, the **Cross Product of sets** A and B is defined as the set of all ordered pairs (x, y) such that x belongs to A and y belongs to B. For example, if A = {1, 2} and B = {3, 4, 5}, then the Caross Product of A and B is {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}. **Functions** are mathematical entities that give unique outputs to particular inputs. There are many ways to think about functions, but they, at all times, have three most important parts: ▪ Input ▪ Relationship ▪ Output **Inductive reasoning** refers to the process of making generalized decisions after observing, and/or witnessing repeated specific instances of something. Conversely, **deductive reasoning** refers to the process of taking the information gathered from general observations and making specific decisions based on that information. **Inductive reasoning is a process of reaching conclusions based on a series of observations while deductive reasoning is a process of reaching conclusions based on previously known facts. Inductive reasoning usually leads to deductive reasoning. A conclusion reached by inductive reasoning may or may not be valid. The conclusions reached by deductive reasoning are correct and valid. Inductive reasoning is used to form hypotheses, while deductive reasoning is used to prove ideas.** A **premise** is a previous statement or proposition from which another is inferred or follows as a conclusion. **Intuition** is the ability to understand something instinctively, without the need for conscious reasoning. A **mathematical proof** is an argument which convinces other people that something is true. Proof is an inferential argument for a mathematical statement. It is a conclusive evidence or an argument that serves to establish a fact or the truth of something. Proof consists of a test or trial of something to establish that it is true. In **direct proof**, the conclusion is established by logically combining the axioms, definitions and earlier theorems. **Certainty** is total continuity and validity of inquiries to the highest degree of precision. Certainty is a conclusion or outcome that is beyond doubt. A mathematical certainty is something that is certain or most likely to happen. Fractions with the same denominators are called **SIMILAR FRACTION** Fractions with different denominators are called **DISSIMILAR FRACTION** A **proper fraction** is a fraction whose numerator is smaller than its denominator.  An **improper fraction** is a fraction whose numerator is equal to or greater than its denominator  GOOD LUCK SA EXAM!!! ©Beyoncé