GEC3-TOPIC-1-AND-2 Mathematics PDF

**What is mathematics?**From the Greek word **"μαθημα"** **(ma ́the ̅ma)** - which means **"that which is learnt",** or "lesson" in modern Greek - derived from **"manthano"** which is equivalent to modern Greek term **"mathaino"** which means **"to learn"Mathematics** is defined as **the study of nu...

**What is mathematics?**From the Greek word **"μαθημα"** **(ma ́the ̅ma)** - which means **"that which is learnt",** or "lesson" in modern Greek - derived from **"manthano"** which is equivalent to modern Greek term **"mathaino"** which means **"to learn"Mathematics** is defined as **the study of numbers** and **arithmetic Operations**Set of tools or a collection of skills that can be applied to questionsOf **"how many"** or **"how much** **Characteristics of Mathematics** Logical SequenceStructurePrecisionGeneralizationAbstractnessMathematical language and symbolsApplicabilityClassification **Nature of Mathematics** A science of measuresAn intellectual gameThe art of drawing conclusionsA tool subjectA system of logical proceduresAn intuitive method **Nature of Mathematics** A set of problem solving toolsAn artLanguageA process of thinkingA study of pattern "Mathematics is the alphabet with which God has written the universe."- Galileo Galillie **PATTERNS AND NUMBERS IN NATURE** **PATTERNS** \- A pattern is an arrangement which helps observers anticipate what they might see or what happens next.- A pattern organizes information so that it becomes more useful.- Mathematics is the study of patterns. That is one reason why those who use patterns to analyze and solve problems often find success compared with those who cannot. Understanding new concepts can also be done in the same way. **Logic pattern can be determined through:**Characteristics of various objects.Order.Patterns that appear in sequence.Patterns that possess similar attributes. **2. Number patterns** - Number patterns such as 2, 4, 6, 810, are familiar to students since they are among the first patterns encountered in school. - The first step in determining the rule that defines the pattern is to look for differences between two consecutive numbers. The number pattern helps make a generalization of how the numbers are arranged in a sequence. If there is no logic (Addition, subtraction, multiplication, division, squares, cubes, primes, etc.) in the differences, find other operations used in pattern - A sequence of numbers that are based on multiplication and division is known as a geometric pattern. - If the numbers in a pattern change in the same way or in the same value each time, then that type of pattern is called a repeating pattern. Example:What is the next number in the sequence: 11, 13, 17, 19, 23, \_\_\_?Answer:If we get the differences between each pair of consecutive numbers, you will get 2, 4, 2 ,4. these differences did not tell us any pattern at all. But notice that the numbers are all consecutive primes. So, the next number must be 29. **4. Geometric patterns** -A geometric pattern is a motif or design that depicts abstract shapes like lines, polygons, and circles, and typically repeats like a wallpaper. A pattern does not need to repeat exactly as long as it provides a way of organizing the artwork.This type of pattern presents objects in a consistent way.Geometric seamless pattern with retro squares 694052 Vector Art at Vecteezy **3. Word patterns**Used in decoding like:**Consonant blend** -- words with a group of two or three consonants that each make its own sound (grow, blend, sleeve, stair, sweet... etc)**Consonant digraph** -- words with two ore three letters come together to create a single sound.(chest, shop, sheep, brush, shirt, shade...)Vowel diphthongs -- vowels that glide in the middle. (boil, soil, brown...)**Vowel Digraph** -- a spelling pattern in which two or more adjoining letters represent a single vowel sound. (school, clean, each, feet, moon, cheese)PATTERNS IN NATURE **1.Symmetry** - The American Heritage Dictionary defined symmetry as an "exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane, or about a center or an axis." In mathematics, an object is said to have symmetry when it remains unchanged after transformations such as rotations and scaling or applied to it. **a. Reflection symmetry**- Also called mirror symmetry or line symmetry, reflection symmetry is made with a line going through an object which divides it into two pieces which are mirror images of each other. Often it is termed as bilateral symmetry, as it divides the object into two mirror images.  **b. Rotation symmetry** - Radial symmetry - The rotation of elements around a common center. -Like reflection symmetry it can occur at any angle or frequency as long as there's a common center around which to rotate.  -used in design to portray motion,speed, and dynamic action. **c. Translational symmetry** - this kind of symmetry is exhibited by objects which do not change its size and shape even if it moved to another location. Note that the movement does not involve rotation. **2. Fractals** -Fractals are never ending patterns that are solved self-similar across different skills. This implies that zooming in the lens on the digital image of the object does not give new details, but only the same as that of the original image. The image just appears. Over and over again no matter how many times the object is magnified.  **3. Spirals** - Spirals are curved patterns made by a series of circular shapes revolving around a central point. Just take fractals. The spiral pattern is very common in nature from the biological molecules that make up organisms to the body plans of certain plants and animals to typhoons and galaxies. Some examples demonstrating the spiral patterns nature or seen and shells of snails, satellite images of a typhoon and horns of a ram. Picture of a spiral staircase 4\. Chaos - chaotic patterns are simple patterns created from complicated underlying behavior. In contrast to popular definitions which relate it to complete disorder, a chaotic pattern is used to describe a kind of order which lack predictability. - nonlinear and the unpredictable - Examples of chaotic patterns in nature can be seen in Vortex street of Clouds and Shells of a mollusk.  **Fibonacci Numbers-** He wrote Liber Abaci -Introduced to Europe by Leonardo Pisano Bigollo/ Leonardo Bonacci/ Leonardo of Pisa/-He introduced Hindu --Arabic numeral system into Europe-Fibonacci is a short term for "Filius Bonacci" which means"the son of Bonacci" Who was Fibonacci? **Golden Ratio** 1, 1, 2 ,3 ,5, 8, 13,...The golden ratio (symbol is the Greek letter \"phi)\ is a special number approximately equal to 1.618It said that if we take 2 consecutive numbers from the Fibonacci numbers their ratio is very close to the Golden ratio(Note: Let "a" be the smaller number and "b" be the larger number.)Use the formula b/aExample:(2,3) = 3/2 = 1.5(3,5) = 5/3 = 1.6666666...(5,8) = 8/5 = 1.6  **Sets and Basic Operations on Sets** What a Set? A collection of -- defined objects or elements. May be mathematical (e.g., numbers and functions) or not. **Finite and infinite sets** **Finite sets**Sets that have a limit. **Infinite sets**- Sets that has no limit. **Universal set and subset** **Universal set** a set that contains all the elements or objects of other sets, including its own elements. **Subset** A set whose members are all contained in another set. **note!** **∅Null set/empty set∊Element of a set∉Not an element of a set⊆Subset** **Venn Diagram** Used to show logical relations between finite sets.  **Operation on sets** Union of a set Elements that consist of both the elements in the involved sets**Intersection of a set** Common elements for both involve sets. FUNCTION **WHAT IS FUNCTION?** A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. **NOTE** All function is a relation, but not all relation is a function. **BINARY OPERATION** -A binary operation is a n operation that takes two input elements from a set and gives a unique result that also belongs to the same set. -A mapping from a set A to a set B is a set of ordered pairs (a, b) where a is an element of A and b is an element of B.-A binary operation on a set S IS a mapping denoted by \* which assigns to each ordered pair of element of S uniquely determined element of S. the set S is said to be closed under the operation \* which means taking the binary operation with two elements of S will give a result that belongs to S.Example:1. Under the set of ℝ, is the operation subtraction in the set ℝ closed?2. The operation \* defined by a\*b=a/b on a set if rational numbers Q except 0.3. The operation \* defined by a\*b=+√ab on a set if rational numbers R.