CHAPTER 3
### Chapter 3 - The Mathematical function and its Graph

#### Section 3.1 - The Scientific Method

Science is a rational method
of observing situations and then forming hypotheses to understand them. The
language of science is mathematics and it is through the mathematical function that a
situation can
be expressed in terms of the conditions that define it. The function is thus a
fundamental concept in mathematics and Calculus is the branch of mathematics
dealing with functions whose dimensions change. What then is a function and how
is it expressed mathematically?

Before we can learn what a function is we need to
understand how mathematics
relates to science. The ** scientific method** is an orderly and efficient way of
analyzing physical situations. Its counterpoint, the **trial and error**
**method**, is something you are
probably more familiar with. Human nature inclines the mind more toward
exploration and experimentation rather than systematic analysis. The trial and
error method reflects the tangents that human logic takes when curiosity
beckons.

The scientific method, on the other hand, provides an
orderly path for the mind to follow in its passage from bewilderment to
enlightenment. Without following the scientific method, an analysis will remain
consistently complex and any theories formed will be too general to explain the
phenomena accurately. Therefore an understanding of the scientific method is
crucial to your understanding of mathematics and development as a scientist.
Here are its main steps.

- Identify the problem or situation
- Narrow down the problem statement while being as
specific as possible.
- Remove external factors to leave behind a few interacting conditions.
- Analyze remaining conditions along with their properties with respect to the
entire system.
- Make reasonable assumptions about the controlled situation, such as mass/ energy/
people balance etc.
- Understand how the conditions come together to define the situation.
- Determine a logical relationship among the conditions.
- Select the most efficient solution that accurately explains the system through the
interacting conditions.

Engineering and science both follow the same systematic
analysis to solve problems and understand the world. Science deals more with
objective and idealized situations while engineering is often concerned with
subjective situations that require sound judgment. Situations are not always
defined by fixed quantifiable conditions. The importance of the scientific
method is its emphasis on clear, logical, and focused thinking that can analyze
a specific situation in relation to the larger system that it is part of. The
method is the quickest cure for frustration and desperation.

Mathematics is a reflection of the orderly analysis and
conclusions of the scientific method. It is the language of science used to
communicate ideas, theories and observations in a concise manner. From the
scientific method, notice how the word, condition, appeared often. In
mathematics conditions are referred to as dimensions. The following section will
take a closer look at what a dimension represents.

**Next section ->**
*
Section 3.2 - Nonsense Detection - The UnScientific Method *