Mathematics in the Modern World
Sunflower
any plant of the genus Helianthus having large flower heads with dark disk florets and showy yellow rays
Fibonacci sequence
Each number is the sum of the two numbers before it
Conventions in the Mathematical Language
** Expressions have different names Example: 5 2 + 3 10÷2 (6 - 2) + 1 1 + 1+ 1 +1+ 1 ** Common in solving expressions is to SIMPLIFY In mathematics, we frequently need to work with numbers, these numbers are the most common mathematical expressions. And, numbers have lots of different names. For example: This simple idea- that numbers has lots of different names - is extremely important in mathematics. This is the same concept as synonyms in English ( words that have the same (or nearly the same) meaning. COMMON IN SOLVING EXPRESSIONS IS TO SIMPLIFY The most common type of problem involving expressions is to Simplify.
D. Mathematics helps predict the behavior of nature and the world
** Mathematics help predict the location, size and timing of natural disasters ** Made possible by the study of fractals. A fractal is a mathematical formula of a pattern that repeats over a wide range of size and time scales. These patterns are hidden within more complex systems. ** Benoit Mandelbrot is the father of fractals, who described how he has been using fractals to find order within the complex systems in nature, such as the shape of coastlines.
F. Mathematics has numerous applications in the world making it indispensable
** Mathematics helps you build things ** Mathematics is helpful in managing financial matters ** Many more...
C. Mathematics helps organize patterns and regularities in the world
** Patterns have underlying mathematical structures ** Every living or nonliving thing in the world may seem to follow a certain pattern on their own. ** The mystery of Fibonacci sequence and the golden ratio as common patterns in nature.
Relations
- A relation is a correspondence between two things or quantities. It is a set of ordered pairs such that the set of all first coordinates of the ordered pairs is called Domain and the set of all the second coordinates of the ordered pairs is called Range. - A relation maybe expressed as a statement, arrow diagram, table, equation, set-builder notation and graph. - Example: R= {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}
What does SIMPLER mean?
- Fewer symbols - Fewer operations - Better suited to current use - Preferred/ style/format 1.Ex: 3 + 1 + 5 and 9 are both names for the same number but 9 uses fewer symbols. 2.Ex. 3 + 3+ 3 + 3 + 3 and 5. 3 are both names for the same number, but 5.3 uses fewer operation 3.Ex. (3.25 units vs 3 1/4 units) 4.(fraction in simplest form is necessary. Example we write ½ instead of 13/26. We usually write reduced form or simplest form
Ideas regarding Mathematical sentence
- Mathematical Sentences have verbs and connectives - Truth of Sentences The notion of truth (the property of being true or false) is of fundamental importance in the mathematical language.
Objects that we use in Math
- Numbers (4 operations and properties) - Variables - Operations (unary & binary) FOUR BASIC CONCEPTS: - Sets (relationships, operations, properties) - Relations (Equivalence relations) - Functions ( injective, Surjective , Bijective) - Binary Operations The students are familiarized with the structure of math) To better understand mathematical language, one must have an understanding of at least a few of the four basic mathematical concepts. injective is one-to-one
How to decide whether something is a Sentence?
- Read it aloud, and ask yourself the question: Does it state a complete thought? If YES, then it is a sentence. - You may also ask yourself the question: Does it make sense to ask about the truth of it?
Some difficulties in math language
- The word "is" could mean equality, inequality or membership in a set - Different uses of a number; to express quantity (cardinal), to indicate the order (ordinal), and as a label (nominal) - Mathematical objects may be represented in many ways, such as sets and functions - The words "and' & "or" means different from its English use
A. Characteristics
1. Precise 2. Concise 3. Powerful
Examples of Patterns and Numbers in Nature and World
1. The snowflake 2. The honeycomb 3. The sunflower 4. The snail's shell 5. Flower's petals 6. Weather
Types of Relations
1. one - to - one relation 2. one - to - many relation 3. many - to - one relation 4. many-to-many relation
Benoit Mandelbrot
A Polish Born, French and American mathematician. He was recognized for his contribution to the field of fractal geometry, which included coining the word "factal" as well as developing a "theory of roughness" and "self similarity in nature." He later discovered the Mandelbrot set of intricate, never ending fractal shapes.
Operations (Unary or Binary)
A Unary operation is an operation on a single element. Example: negative of 5 multiplicative inverse of 7 - A binary operation is an operation that combines two elements of a set to give a single element. e.g. multiplication 3 x 4 = 12
Binary Operations
A binary operation on a set A is a function that takes pairs of elements of A and produces elements of A from them. We use the symbol * to denote arbitrary binary operation on a set A. Four Properties: 1.Commutative x* y = y *x 2.Associative x* (y*z) = (x*y)* z 3.Identity e*x = x *e 4.Inverse x*y = y*x = e In other words, It is a function with the set all pairs (x,y) of elements of A as its domain and with A as its range.
Variable
A factor that can change in an experiment A variable is any letter used to stand for a mathematical object. (in this manner, the students are familiarized with the structure of math) Variables: Suppose we say something like" At time t the speed of the projectile is v. The letters t and v stand for real numbers and they are called VARIABLES. More generally, a variable is any letter used to stand for a mathematical object, whether or not one thinks of that object as changing through time. injective is one-to-one
Functions
A function is a relation such that each element of the domain is paired with exactly one element of the range. To denote this relationship, we use the functional notation: y = f(x) where f indicates that a function exists between variables x and y. The concept of function provides the essential tool in applying mathematical formulations in solving problems. For instance, the statement "the area of a circle depends on its radius" can be denoted as A= f(r), where A represents the area and r, the radius. This is read as "Area is a function of radius".
mathematical sentence
A mathematical sentence is the analogue of an English sentence; it is a correct assignment of mathematical symbols that states a complete thought.
Terms
A number, a variable, or a product of numbers and variables.
Fibonacci sequence
A sequence of numbers in which each number is the sum of the preceding two.
Nature of Mathematics
A. Patterns and Numbers in Nature and World B. Fibonacci Sequence 2. Exponential Growth Model
Decay Formula
A=P(1-r)^t
Numbers and 4 operations
As you know, people all around the world speak different languages. You're probably even learning another language at school. Did you know that math is also a special kind of language that is common around the world. However, instead of writing sentences with words, we write mathematical sentences with numbers and symbols. Let's start with some common expressions relating to the four operations:
Patterns and Numbers in Nature and World
By using Mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that governs natural processes.
precise
Clearly expressed; exact; accurate in every detail be able to make very fine distinctions
second difference
Differences that are found by subtracting consecutive first differences from one another.
Leonardo of Pisa (Fibonacci)
Famous for sequence of numbers that mimic properties of nature
Relations in Language of Math
Grammatical rules for the use of symbols - To use < in a sentence, one should precede it by a noun and follow it by a noun. - Other examples of relations are "equals" and " is an element of" It is important when specifying a relation to be careful about which objects are to be related.
Patterns and Numbers in Nature and World
Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns called Mathematics.
Noun vs Sentence
In English , nouns are used to name things we want to talk about (like people, places and things); whereas sentences are used to state complete thoughts. A typical English sentences has at least one noun, and at least one verb. For example, Gemma loves Mathematics.
Liber Abaci
In this book Fibonacci explained why the Hindu-Arabic numeration system that he had learned about during his travels was a more sophisticated and efficient system than the Roman numeration system.
The Grammar of Mathematics
It is the structural rules governing the use of symbols representing mathematical objects. Express the following using mathematical symbols a. 5 is the square root of 25 b. 5 is less than 10 c. 5 is a prime number The main reason for the importance of mathematical grammar is that statements of mathematics are supposed to be precise. Mathematical sentences become highly complex if the parts that made them up were not clear and simple which makes it difficult to understand.
Language of Mathematics
Like any language, Mathematics has its own symbols, syntax and rules. - to understand the expressed ideas - to communicate ideas to others
Chapter 2
Mathematical Language and Symbols
Mathematical reasoning
Mathematical reasoning refers to the ability of a person to analyze problem situations and construct logical arguments to justify his process or hypothesis, to create both conceptual foundations and connections, in order for him to be able to process available information.
Mathematics
Mathematics is about ideas -- relationships, quantities, processes, ways of figuring out certain kinds of things, reasoning, and so on. It uses words. When we have ideas, we often want to talk about them; that is when we need words. Words help us communicate. The language of mathematics can be learned, but requires the efforts needed to learn any foreign language. Thus, we need to get extensive practice with mathematical language ideas, to enhance the ability to correctly read, write, speak, and understand mathematics.
Exponential Growth Model
Model's rate of change increases exponentially
Chapter 1
Nature of Mathematics
A. Patterns and Numbers in Nature and World
Patterns are regular, repeated, or recurring forms or designs which are commonly observed in natural objects such as six-fold symmetry of snowflakes, hexagonal structure, and formation of honeycombs, tiger's stripes and hyena's spots. Humans are hard wired to recognize patterns and by studying them we discovered the underlying mathematical principles behind nature's designs.
Benoit Mandelbrot
Polish-French mathematician who formulated fractal theory
Exponential Growth Model
Population growth and bacterial decay can be modeled by the exponential growth or decay formula.
Chapter 3
Problem Solving and Reasoning
Ideas regarding Mathematical sentence
SENTENCES HAVE VERBS and CONNECTIVES Just as English sentences have verbs, so do mathematical sentences. In the mathematical sentence 3+4= 7. The equal sign is actually the verb and indeed one of the most popular mathematical verbs. The symbol "+" in 3 + 4 = 7 is a connective which is used to connect objects of a given type TRUTH OF SENTENCES: Sentences can be true or false. It makes sense to ask the TRUTH of a sentence. Ask IS IT TRUE? IS IT FALSE? IS IT SOMETIMES TRUE? SOMETIMES FALSE?
Fractal Geometry
Study of fragmented geometric irregular shapes, such as those in lining of intestine
Binet's formula
The advantage of this formula over the recursive formula 𝐅𝐧 = 𝐅𝐧−𝟏 + 𝐅𝐧−𝟐 is that you can determine the nth Fibonacci number without finding the two preceding Fibonacci numbers.
Weather
The condition of Earth's atmosphere at a particular time and place.
Evaluating Functions
The functional notation y = f(x) allows us to denote specific values of a function. To evaluate a function is to substitute the specified values of the independent variable in the formula and simplify. Example: When f(x) = 2x - 3, find f(2) Solution: f(2) = 2(2) - 3 = 4 - 3 f(2) = 1 One of the most basic activities in mathematics is to take a mathematical object and transform into another one.
Inverse of a Function
The inverse of a function is another function that that undoes it, and that it undoes. For example, the function that takes a number n to n - 5 is the inverse of the function that takes n to n + 5. What is the inverse of y = 2x? One of the most basic activities in mathematics is to take a mathematical object and transform into another one.
Functions
The notation f : A→B is used to denote a function which means that f is a function with domain A and range B; f(x) = y means that f transform x (which must be an element of A) into y ( which must be an element of B)
Terms
The numbers in a sequence that are separated by commas.
first differences
The values determined by subtracting consecutive y-values in a table when the x-values are consecutive integers.
Fibonacci sequence
This sequence is formed by adding the preceding two numbers, starting with 0 and 1. The most aesthetically pleasing proportion of the Golden ratio is approximated by the ratios of Fibonacci numbers.
B. Expressions vs Sentences
We call the mathematical analogue of noun as EXPRESSION. Thus an expression is a name given to a mathematical object of interest. Whereas, in English we need to talk about people, places, and things, we know that in mathematics has much different objects of interest.
Snowflake
a flake of snow, especially a feathery ice crystal, typically displaying delicate sixfold symmetry.
nth term of a sequence
a formula for the general term of a sequence an=a1+(n-1)d
one-to-one function
a function where each element of the range is paired with exactly one element of the domain
Sets
a group of repetitions a collection of objects, and in mathematical discourse these objects are mathematical ones such as numbers, points in space or other sets. Sets are very useful if one is trying to prove statements for this helps a lot if one can devise a very simple language - with a small vocabulary and an uncomplicated grammar- into which it is in principle possible to translate all mathematical arguments. For example with the help of a membership symbol ∈ such as 5 is a Natural number can just simply written as 5∈ N.
mathematical sentence
a math equation or expression with numbers, operations, symbols, or variables
Exponential Growth Model
a mathematical description of idealized, unregulated population growth
Recursive definition for a sequence
a sequence defined by giving the first term (or the first few terms) along with a procedure for finding the subsequent terms
first differences
a sequence formed by subtracting each term of a sequence from the next term
Fibonacci sequence
a sequence of numbers in which each number equals the sum of the two preceding numbers
Problem
a situation that conforms the learner, that requires resolution, and for which the path of the answer is not immediately known. There is an obstacle that prevents one from setting a clear path to the answer.
mathematical sentence
a statement that indicates a relationship between two mathematical expressions
Honeycomb
a structure of rows of hexagonal wax cells, formed by bees in their hive for the storage of honey, pollen, and their eggs.
nth term of a sequence
a1 represents the fi rst term of a sequence. a2 represents the second term of a sequence. a3 represents the third term of a sequence. . . . an represents the nth term of a sequence
Sequence
an ordered list of numbers called terms that may have repeated values. The arrangement of these terms is set by definite rule. Applying the rule to the previous terms of the sequence generates the different terms of the sequence. Example: 5, 14, 27, 44, 65, ... In the above sequence, 5 is the fi rst term, 14 is the second term, 27 is the third term, 44 is the fourth term, and 65 is the fi fth term.
poweful
be able to express complex thoughts with relative ease
concise
brief and to the point use symbols to be able to express more
nth term of a sequence
formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next.
Exponential Growth Model
growth model that estimates a population's future size after a period of time based on the intrinsic growth rate and the number of reproducing individuals currently in the population
Problem Solving
has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills (Goldstein & Levin, 1987).
Three dots "..."
indicate that the sequence continues beyond 65, which is the last written term. It is customary to use the subscript notation an to designate the nth term of a sequence.
Recursive definition for a sequence
is one in which each successive term of the sequence is defined by using some of the preceding terms. If we use the mathematical notation Fn to represent the nth Fibonacci number, then the numbers in the Fibonacci sequence are given by the following recursive definition.
Language of Mathematics
makes it easy to express the kinds of thoughts that mathematicians like to express. It is, precise... concise... and powerful.... Concise means stating something succinctly, using as few words as possible yet still conveying the full meaning. Precise means exact, accurate. It is often used in mathematical or scientific contexts in which definite, fixed statements or measurements are demanded.
Leonardo of Pisa (Fibonacci)
one of the best-known mathematicians of medieval Europe. In 1202, after a trip that took him to several Arab and Eastern countries, Fibonacci wrote the book Liber Abaci
Difference table
shows the differences between successive terms of the sequence
Fractal Geometry
the study of surfaces with a seemingly infinite area, such as the lining of the small intestine
Fractals
the type of geometry that creates broken patterns out of a smaller version of a design
Fibonacci sequence
𝐅𝟏 = 𝟏, 𝐅𝟐 = 𝟏, 𝐭𝐡𝐞𝐧 𝐅𝐧 = 𝐅𝐧−𝟏 + 𝐅𝐧−𝟐 𝐟𝐨𝐫 𝐧 ≥ 𝟑.
