Lesson 1: Nature of Mathematics
Like symmetry, mathematical sequences is
another concept that shows up in nature. The
Fibonacci sequence involves adding the two
previous numbers in the sequence to arrive at the
next number i.e.: 1,2,3,5,8, etc. Intriguingly, this
sequence is often found in nature and is
frequently called the golden ratio. The number of
petals on flowers is usually a Fibonacci number.
In the general sense of the word, patterns are
regular repeated recurring forms or designs.
Patterns in nature are visible regularities of form
found in the natural world. These patterns recur
in different contexts and can sometimes be
modelled mathematically. Natural patterns
include symmetries, trees, spirals, meanders,
waves, foams, tessellations, cracks and stripes.
Fibonacci sequence has many interesting
properties. Among these is that this pattern is
very visible in nature. Some of natures' most
beautiful patterns, like spiral arrangements of
sunflowers, the number of petals in a flower, and
the shape of snail's shell– things that we looked
at earlier in this chapter– all contain Fibonacci
numbers. It is also interesting to note that the
ratios of successive Fibonacci numbers approach
the number (Phi), also known as the Golden
Ratio. This approximately equal to 1.618.
Number Patterns
Number pattern is a pattern or sequence in a
series of numbers. This pattern generally
establishes a common relationship between all
numbers.
Fibonacci Sequence
It is named after the Italian mathematician
Leonardo of Pisa, who was better known by his
nickname Fibonacci. He is said to discover this
sequence as he looked at how a hypothesized
group of rabbits bred and reproduced. The
problem involved having a single pair of rabbits
and then finding out how many pair of rabbits
will be born in a year, with the assumption that a
new pair of rabbits is born each month and this
new pair, in turn, gives birth to additional pairs
of rabbits beginning at two months after they
were born. He noted that the set of numbers
generated from this problem could be extended
by getting the sum of the two previous terms.
1, 1, 2, 3, 5, 8, 13,…
Mathematics for our World
Language of Mathematics
Language is very powerful. Language is very
powerful. It is used to express our emotions,
thoughts, and ideas. However, if the recipient of
the message cannot understand you, then there is
no communication at all. It is very important that
both of you understand the language.
Mathematics is very hard for others to study
because they are very overwhelmed with the
numbers, operations, symbols and formulas. On
the other hand, of one knows how to interpret and
understand these things, then the subject will be
comprehensible.
Comprehending a message is better
understood once a person understand how things
are said and may know why it is said. The use of
language in mathematics is far from ordinary
speech. It can be learned but needs a lot of effort
like learning a new dialect or language. The
following are the characteristics of a language of
mathematics: precise, concise, and powerful.
1. The language of Mathematics is Precise.
This characteristic will be able to make a very
fine distinction. And an able to clearly
communicate and ask question as solving a
problem, this also expand mathematics
vocabulary and build capacity to define and
learning new things.
Example:
The value of pi is 3.14159265359 approximately.
A number which is quite accurate but not precise
is 3.141.
2. The language of Mathematics is Concise
This characteristic will be able to say things
briefly and easier to critique. In this
characteristic involves using the most effective
words in order to get one’s point across and
entails using a minimal amount of effective word
to make one’s point.
8 ∙ y =8y
a ∙ b ∙c =abc
t ∙ s ∙ 9= 9st
It is conventional to write the number first
before the letters. If in case the letters are more
than one, you have to arrange the letters
alphabetically.
Sets are usually represented by uppercase
letters like S. the symbols R∧N represents the
real numbers and the set of natural numbers,
respectively. A lowercase letter near the end of
the alphabet like x, y, or z represents an element
of the set of real numbers.
Numbers and Terminologies
Example:
The Language of Set
The solution should be clear and complete.
Use of the word set as a formal mathematical
term was introduced in 1879 by Georg Cantor
(1845–1918). For most mathematical purposes
we can think of a set intuitively, as Cantor did,
simply as a collection of elements.
3. The language of Mathematics is Powerful.
In this characteristic will be able to express
complex thoughts with relative ease. It gives a
way to understand patterns and to quantify
relationships. Using this make sense of the world
and solve complex and real problems.
Expressions Versus Sentences
For instance, if C is the set of all countries that
are currently in the United Nations, then the
United States is an element of C, and if I is the
set of all integers from 1 to 100, then the number
57 is an element of I.
You learned in your English subject that
expressions do not state a complete though, but
sentences do. Mathematical sentences state a
complete thought. On the other hand,
mathematical expressions do not. You cannot
test if it is true or false.
Example 1 – Using the Set-Roster Notation
Mathematical Convention
The common symbol used for multiplication is x
but it can be mistakenly taken as variable x.
There are instances when the centered dot(.) is a
shorthand to be used for multiplication especially
when variables are involved. If there will be no
confusion the symbol may be dropped.
a. Let A = {1, 2, 3}, B = {3, 1, 2}, and C = {1,
1, 2, 3, 3, 3}. What are the elements of A, B, and
C? How are A, B, and C related?
b. Is {0} = 0?
c. How many elements are in the set {1, {1}}?
d. For each nonnegative integer n, let Un = {n, –
n}. Find U1, U2, and U0.
Certain sets of numbers are so frequently
referred to that they are given special symbolic
names. These are summarized in the following
table:
Another way to specify a set uses what is
called the set-builder notation.
Example 2 – Using the Set-Builder Notation
The set of real numbers is usually pictured as
the set of all points on a line, as shown below.
The number 0 corresponds to a middle point,
called the origin. A unit of distance is marked off,
and each point to the right of the origin
corresponds to a positive real number found by
computing its distance from the origin.
Given that R denotes the set of all real
numbers, Z the set of all integers, and Z+ the set
of all positive integers, describe each of the
following sets.
Each point to the left of the origin
corresponds to a negative real number, which is
denoted by computing its distance from the
origin and putting a minus sign in front of the
resulting number.
SUBSETS
The set of real numbers is therefore divided
into three parts: the set of positive real numbers,
the set of negative real numbers, and the number
0.
Note that 0 is neither positive nor negative.
Labels are given for a few real numbers
corresponding to points on the line shown below.
The real number line is called continuous
because it is imagined to have no holes. The set
of integers corresponds to a collection of points
located at fixed intervals along the real number
line.
A basic relation between sets is that of subset.
It follows from the definition of subset that
for a set A not to be a subset of a set B means that
there is at least one element of A that is not an
element of B.
Symbolically:
Example 4 – Distinction between ∈ and ⊆
Which of the following are true statements?
Thus, every integer is a real number, and
because the integers are all separated from each
other, the set of integers is called discrete. The
name discrete mathematics comes from the
distinction between continuous and discrete
mathematical objects.
a. 2 ∈ {1, 2, 3}
b. {2} ∈ {1, 2, 3}
c. 2 ⊆ {1, 2, 3}
d. {2} ⊆ {1, 2, 3}
e. {2} ⊆ {{1}, {2}} f. {2} ∈ {{1}, {2}}
CARTESIAN PRODUCTS
a. Is (1, 2) = (2, 1)?
b. Is
?
c. What is the first element of (1, 1)?
Example 6 – Cartesian Products
Let A = {1, 2, 3} and B = {u, v}.
a. Find A × B
b. Find B × A
c. Find B × B
d. How many elements are in A × B, B × A, and
B × B?
e. Let R denote the set of all real numbers.
Describe R × R.