MATH 101 REVIEWER Flashcards
Characteristics of the Math language
Precise, Concise, Powerful
It creates a complex schedules for sports tournaments, making sure teams have equal numbers of home and away games, and do not travel for too long
Mathematics
Why is language important?
To understand expressed ideas, to communicate ideas
Objects that we use in Math
-Numbers (operations and properties)
-Variables (Free and Bound)
-Operations (Unary and Binary)
-Sets (Relationships, Operations, Properties)
-Relations (Equivalence Relation, Functions)
-Functions (Injective, Surjective, Bijective)
it is changing the grouping of the numbers in addition or multiplication will not change the result
Associative Property
In other words adding zero to a number does not change its value
Additive Identity Property
Properties of Real Numbers
Closure, Commutative, Associative, Distributive, Identity + x, Inverse + x
There exists a unique number 1 such that the number 1 preserves identities under multiplication.
Multiplicative Identity Property
For each real number a there exists a unique real number -a such that their sum is zero
Additive Inverse Property
For each real number, a there exists a unique real number 1 over a such that their product is 1
Multiplicative Inverse Property
It is changing the order of the numbers in addition or multiplication will not change the result
Commutative Property
Focused on the “structure”, Structural rules governing.
Grammar of Mathematics
explains the
universe, such as why bees have
hexagonal honeycombs, and how
many galaxies there are.
Mathematics
In other words multiplying a number by 1 does not change the value of the number.
Multiplicative Identity Property
There exists unique number 0 such that zero preserves identities under addition
Additive Identity Property
Multiplication distributes over addition
Distributive Property
words opposites add to zero
Additive Inverse Property
states: a + (b +c) = (a + b) + c
Associative Property
it powers graphics software, so that you can draw perfect curves and save pictures digitally.
Mathematics
A mathematician, mathematics is a formal system of thought for recognizing, classifying, and exploiting patters.
Ian Stewart
Philosopher: Mathematics is the science of quantity, Philosopher and Polymath
Aristotle
is the study of numbers, quantities, shapes, and patterns. It involves logical reasoning, problem-solving, and the ability to analyze and interpret data. it also provides the tools and frameworks to understand and describe the world around you.
Mathematics
The Science of Indirect Measurements, A mathematician
Auguste Cosme
Mathematics teaches logical reasoning, critical thinking, and solving skills. These skills are essential in everyday life for making decisions, analyzing situations, and finding solutions to complex problems.
Problem Solving skills
Mathematics is the language in which god has written in the universe. Phycist, engineer, scientist, mathematician, polymath.
Galileo Galilei
is a fundamental discipline that plays a critical role in various aspects of life, science, and technology.
Mathematics
A mathematician, mathematics is the classification and study of all possible patterns and relationships.
Walter Warwick Sawyer
3 Core Concept of Mathematics:
Numbers: The foundation of mathematics, including natural numbers, integers, fractions, and real numbers.
Operations: Addition, subtraction, multiplication, division, and more complex operations.
Patterns and Relationships: Identifying and analyzing patterns to understand relationships between numbers and variables.
Mathematics is essential in daily activities such as shopping, cooking, home budgeting, and time management. Understanding basic mathematical concepts helps people navigate everyday tasks more efficiently.
Supportive Daily Life
Mathematician, Mathematics is our one and only strategy for understanding the complexity of nature.
Ralph Abraham
Are symbols or letters used to represent numbers or other mathematical objects. They are essential in expressing general mathematical statements, equations, and functions, allowing for the formulation of relationships and problem-solving strategies.
Variables
Many modern technologies, including computers, telecommunications, artificial intelligence, and cryptography, are built on mathematical theories. Innovations in these fields often require advanced mathematical understanding.
Innovation and Technology
it’s not just about numbers, it is a way of thinking, a method of problem-solving, and a tool for understanding the world.
Mathematics
enable generalization in mathematics, allowing for the formulation of equations, functions, and models that can apply to many different situations.
Variables
Learning mathematics stimulates _____development, enhancing memory, attention, and analytical abilities. It fosters an ability to think abstractly and systematically, which is beneficial in various intellectual pursuits.
Cognitive Development
A value that does not change within a given context or equation.
Constant Variable
Types of Variables:
Independent Variable: The variable that is manipulated or chosen in an equation or function. It represents the input or cause.
Dependent Variable: The variable whose value depends on the independent variable. It represents the output or effects.
Constant: A value that does not change within a given context or equation.
Parameter: A variable that remains constant within a specific context but can change when the context changes.
Random Variable: A variable that takes on different values based on the outcomes of a random process.
Discrete Variable: A variable that can take on a finite or countable number of values.
Continuous Variable: A variable that can take on any value within a given range, often involving decimals.
The variable whose value depends on the independent variable. It represents the output or effects
Dependent Variable
it provides a formal way to describe collections of objects and their relationships.
Language of sets
The variable that is manipulated or chosen in an equation or function. It represents the input or cause.
Independent Variable
underpins
statistical analysis which enables
medical, biological, psychological
and other research.
Mathematics
provides a way to describe and analyze how different sets of elements are related to each other.
language of relations and functions
Types of Relation:
Reflexive: Each element is related to itself.
Symmetric: If one element is related to another, the reverse is true.
Transitive: If one element is related to a second, and the second to a third, the first is related to the third.
A variable that remains constant within a specific context but can change when the context changes.
Parameter Variable
A variable that can take on any value within a given range, often involving decimals.
Continuous Variable
A ____ is a way to describe a relationship between elements of two (or more) sets.
Relations
A variable that can take on a finite or countable number of values.
Discrete Variable
powers search
engines so that you can be sent
to the most popular hits within
seconds.
Mathematics
Is the set of all x or input values. We may describe it as the collection of the first values in the ordered pairs.
Domain
A set is a collection of distinct objects considered as a whole.
Language of sets
If one element is related to another, the reverse is true.
Symmetric Relation
A variables that takes on different values based on the outcomes of a random process.
Random Variable
If one element is related to a second, and the second to a third, the first is related to the third.
Transitive Relation
is simply a set or collection of ordered pairs, Nothing really special about it. An ordered pair, commonly known as a point, has two components which are the x and y coordinates.
Relation
it drives
spreadsheets, which host millions
of models of businesses, projects
and such.
Mathematics
is the set of all Y or output values. We may describe it as the collection of the second values in the ordered pairs.
Range
Each element is related to itself.
Reflexive
makes calculators,
which means you don’t need to
waste brain power on
calculations.
Mathematics
it is a “special” kind of relation because it follows an extra rule. Just like a relation, a ____ is also a set of ordered pairs; however, every x-value must be associated to only one y-value.
function
is when a shape looks identical to its original shape after being flipped or turned.
Symmetry
is a fascinating topic that highlights the intrinsic connection between mathematics and the natural world. Mathematics is not just a tool for understanding the universe but is deeply embedded in the structures and patters we observe in nature.
Patterns and Numbers in Nature and the World
Are linear openings that form in materials to relieve stress. The pattern of ____ indicates whether the material is elastic or not. Some examples are old pottery surface, drying inelastic mud, and palm trunk with branching vertical ___.
Cracks
Two main types of symmetry:
Reflective, Rotational
is when an object exhibits self-similar shape or form at any scale and repeat itself overtime. Trees are natural Fractals, patterns that repeat smaller and smaller copies of themselves to create the biodiversity of a forest.
Fractal Pattern
allows us to weigh
up the trade-off between over forecasting tornadoes and failing
to warn.
Mathematics
is an organized arrangement of objects in space or time., It must have something that is repeated either exactly or according to recognizable transformations.
Pattern
The hexagonal pattern in honeycomb structures is an efficient way of using space and resources, minimizing the amount of wax needed to build the hive.
Bees and Hexagons
means that one half of an image is the mirror image of the other half (think of a butterfly’s wings).
Reflective, or line symmetry
Forms a class of patterns found in nature. The arrays of hexagonal cells in a honeycomb or the diamond-shaped scales that pattern snake-skin are natural examples of what?
Tesselation or Tiling Pattern
is the greatest European mathematician of middle ages.
Fibonacci
means that the object or image can be turned around a center point and match itself some number of times (as in a five-pointed star).
Rotational symmetry
can be observed in many things around us.
Patterns
Animals, such as birds and fish, often follow complex patterns during ___, which can be described mathematically using vectors and forces. (Patterns in Animal Behavior)
Migration and Navigation
Known as Fibonacci
Leonardo of Pisa
Date of Birth and Death of Leonardo of Pisa
Born in 1170 and died in 1240.
a book that introduced to the western world
Liber Abaci (The Book of Calculation) published in 1202
is an integer in the infinite sequence, 1,1,2,3,5,8,13 … of which the first two terms are 1 and 1 and each succeeding term is the sum of the two immediately preceding.
Fibonacci Sequence
reduces waste
when used for inventory control,
for distribution networks, for
product creation.
Mathematics
He introduced the Arabic number system in Europe
Leonardo of Pisa - Fibonacci
Fibonacci presented a problem about rabbit population growth, which led to the sequence that become famous.
Fibonacci’s Problem
