GE 04 - Mathematics in the Modern World

Mathematics?

• the science of structure, order, and relations that has evolved from elemental practices of counting, measuring, describing the shapes and characteristics of objects

• its a human activity that involves observing, representing and investigating patterns, and qualitative relationships

What math helps?

• It helps to develop mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute to decision-making.

_⁽⁵⁾ Mathematics is an/a…

• arts

• language

• process of thinking

• study of patterns

• set of problem solving tools

_⁽⁶⁾ Nature of Mathematics

• a science of measure

• an intellectual game

• an intuitive method

• a tool subject

• the art of drawing conclusions

• a system of logical procedure

I. Mathematics for our World _⁽⁷⁾

• Medical Field

• Agriculture

• Political Science

• Psychology

• Accounting

• Economics

• Archaeology

Patterns

• is a repeated design or recurring sequence

• Is the repeated or regular way in which something happens or is done

• any regularly repeated arrangement, especially a design made from repeated lines, shapes, or colors on a surface

_⁽⁶⁾ Different kind of Patterns

1. Patterns of Visuals

2. Patterns of Flow

3. Patterns of Movement

4. Patterns of Rhythm

5. Patterns of Texture

6. Geometric Patterns

Patterns of Visuals

• are often unpredictable, never quite

  repeatable, and often contain fractals.

ex: flower petals, fractals, trees

Patterns of Flow

• The flow of liquids provides an inexhaustible supply of nature’s patterns.

• Patterns of flow are usually found in the water, stone, and even in the growth of trees.

ex: rivers, ocean waves

Patterns of Movement

• In the human walk, the feet strike the ground in a regular rhythm: (e,g walking) the left-right-left-right-left rhythm.

Patterns of Rhythm

• conceivably the most basic pattern in nature.

ex: heartbeat, lungs, pulse

Patterns of Texture

• a quality of a certain object that we sense through touch.

• It exists as a literal surface that we can feel, see, and imagine.

• Textures are of many kinds. It can be bristly, and rough, but it can also be smooth, cold, and hard.

ex: cracks,

Geometric Patterns

• a kind of pattern which consists of a series of shapes that are typically repeated.

ex:

Symmetries

• a figure can be folded or divided into two with two halves which are the same, such figure is called a symmetric figure.

Reflection Symmetry

sometimes called line symmetry or mirror symmetry, captures symmetries when the left half of a pattern is the same as the right half.

Rotation Symmetry

also known as rotational symmetry, capture symmetries when it still looks the same after some rotation (of less than one full turn).

Translational Symmetry

• is where a figure or an image is translated at a set distance in the same direction as the original.

• The spaces between points, angles, sizes, and shapes of the figure will not change.

• The only thing that changes is its location.

Fractals

• From the word fraction, or part of a whole, fractals are self-similar, iterated mathematical constructs where shrinking and moving are applied many times.

e.g broccoli, tree branches, succulents, leaf veins

Spirals

• a curve which turns around some central point, getting further away, or closer as it goes.

e.g shells, ram horns, milky way galaxy

Spots and Stripes

• Patterns like spots and stripes that are commonly present in different organisms are results of reaction-diffusion system.

• The size and the shape of the pattern depend on how fast the chemicals diffusive and how strongly they interact.

e.g zebra, cheetah

Waves and Dunes

• are disturbances that carry energy as they move. When winds blow over large bodies of sand, they create dunes.

• may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or seif ('sword') shapes.

e.g sand, waves

Cracks

• are linear openings that form in materials to relieve stress. When a material fails in all directions it results in cracks. The patterns created reveal if the material is elastic or not.

Tessallations

• (repeating tile patterns or tilling) is a pattern made up of one or more geometric shapes joined together without overlapping or forming any gaps to cover a plane.

e.g honeycomb, iguana’s scales, turtle shell, dragon fly’s wings

Types of Sequences

1. Arithmetic Sequence

2. Geometric Sequence

3. Harmonic Sequence

4. Fibonacci Sequence

Arithmetic Sequence

• is an ordered set of numbers that have a common difference between each consecutive term.

Formula: an = a1 + n − 1 d

where:

a1 = first term of the sequence

d = common difference

n = term position

an = n th term of the sequence

Geometric Sequence

• are sequences where the term can be determined by multiplying the previous term with a fixed factor we call the common ratio.

Formula: an = a1r n−1

where:

a n = n th term of the sequence

a1= first term of the sequence

r = common ratio

n = term position