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Arithmetic is the study of numbers in action. The four basic operations of arithmetic are addition, subtraction, multiplication, and division.
Fractions, percentages, powers, and roots are developed from these four basic operations. Arithmetic has its own shorthand. For example, the
symbol + means add, the symbol - means take away, the symbol × means multiply by, and the symbol ÷ means divide by.
In the branch of mathematics called algebra, letters are used to stand
for unknown amounts. Some letters are called variables because the numbers
they represent vary within the same equation. Other letters are called
constants because they represent numbers with a fixed value that never
changes.
The branch of mathematics that deals with the relationships between the
sides and angles of triangles is called trigonometry. Provided enough
information is already known, trigonometry can be used to find the
measurements of unknown sides and angles in a triangle.
Geometry is the study of shape, size, and other properties of figures
in space. Planes, squares, spheres, and other geometric figures are
abstract ideas. We can never draw a perfect square, though geometric
techniques can help us to construct an approximation. The principles of
geometry were established by the Greek mathematician Euclid (c.
330-275 BC). More recently, geometry has been developed to include
subjects such as topology. Geometric principles are used in
artistic composition, architecture, navigation, electronic circuit design,
and in other areas of mathematics. Many of the objects we see around us
have been designed using the principles of geometry.
Established by Aristotle 2,000 years ago, logic is the
study of general patterns of reasoning, without reference to particular
meanings or contexts. If an object must be either blue or green, and if it
is not blue, then logic leads us to the conclusion that it must be
green. Logical reasoning from the given premises (initial statements)
cannot reveal what "blue" or "green" mean, or why an
object cannot be both. Logical statements can be formalized using, for
example, Boolean algebra or propositional calculus.
Developments in logic during the 20th century have substantially revised
traditional notions of reasoning.
According to Tweedledee, a character created by the writer Lewis
Carroll, "If it was so, it might be; and if it were so, it would
be; but as it isn't, it ain't. That's logic."
Computer chips contain millions of logic gates connected together.
Each gate is a simple logical switch.
The branch of mathematics concerned with collecting and interpreting
data is called statistics. This data typically consists of
measurements from scientific experiments or the business world.
Qualitative data, representing people's opinions or beliefs, for example,
may also be analyzed statistically. We can use statistics to summarize
large amounts of data visually in graphs and charts. We can also make
general predictions based on limited data, and estimate the accuracy of
these predictions. Unlike the certainties of geometry, results in
statistics are based on probabilities. Statistical analysis remains a
powerful tool as long as we bear this in mind. Statistics also play an
important role in financial forecasting.
Independently developed by Isaac Newton and Gottfried Leibniz
in the 17th century, calculus has had a profound impact on mathematics,
science, economics, and commerce. The fundamental concepts of modern
calculus are limits and change.
Differential calculus studies the rate of change of variables, and gives
the gradient of a curve. Integration, the inverse operation of
differentiation, extends the idea of addition to enable us to find the sum
of continuously changing quantities. The method of integration simplifies
many tasks, such as finding the areas of curved surfaces.
Newton conceived of the Universe as a clockwork mechanism governed by
calculus. Our metaphors for physical reality have now changed, but
calculus remains essential for working out orbits and spacecraft
trajectories. |
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