Utasoma Notes katika mfumo wa PDF
(You will read the Notes in form of PDF)
Click the Chapters below to view the Notes:
Source
TIE (Tanzania Institute of Education)
The Meaning of Mathematics
Mathematics is
the science and study of quality, structure, space, and change. Mathematicians
seek out patterns, formulate new conjectures, and establish truth by rigorous
deduction from appropriately chosen axioms and definitions.
There
is debate over whether mathematical objects such as numbers and points exist
naturally or are human creations. The mathematician Benjamin Peirce called
mathematics “the science that draws necessary conclusions”. Albert
Einstein, on the other hand, stated that “as far as the laws of mathematics
refer to reality, they are not certain; and as far as they are certain, they do
not refer to reality.”
Mathematics is
an area of knowledge that includes the topics of numbers,
formulas and related structures, shapes and the spaces in which they are
contained, and quantities and their changes. These topics are represented in
modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively.
There is no general consensus among mathematicians about a common definition
for their academic discipline.
Most
mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of
either abstractions from nature or in
modern mathematics entities that are stipulated to have certain properties,
called axioms. A proof consists of a succession of
applications of deductive rules to already established results.
These results include previously proved theorems, axioms, and in case of abstraction from nature some
basic properties that are considered true starting points of the theory under
consideration.
Mathematics is
the science that deals with the logic of shape, quantity and arrangement. Math
is all around us, in everything we do. It is the building block for everything
in our daily lives, including mobile devices, computers,
software, architecture (ancient and modern), art, money, engineering and even
sports.
Since
the beginning of recorded history, mathematical discovery has been at the
forefront of every civilized society, and math has been used by even the most
primitive and earliest cultures.
The need for math arose because of the increasingly complex demands from
societies around the world, which required more advanced mathematical
solutions, as outlined by mathematician Raymond L. Wilder in his book “Evolution
of Mathematical Concepts(opens in new tab)” (Dover Publications,
2013).
Through
abstraction and logical reasoning mathematics evolved from counting,
calculation, measurement, and the systematic study of the shapes and motions of
physical objects. Practical mathematics has been a human activity for as far
back as written records exist. Rigorous arguments first appeared in Greek
mathematics, most notably in Euclid’s Elements. Mathematics continued to
develop, in fitful bursts, until the Renaissance, when mathematical innovations
interacted with new scientific discoveries, leading to an acceleration in
research that continues to the present day.
Today,
mathematics is used throughout the world as an essential tool in many fields,
including natural science, engineering, medicine, and the social sciences.
Applied mathematics, the branch of mathematics concerned with application
of mathematical knowledge to other fields, inspires and makes use of new
mathematical discoveries and sometimes leads to the development of entirely new
disciplines. Mathematicians also engage in pure mathematics, or mathematics for
its own sake, without having any application in mind, although practical
applications for what began as pure mathematics are often discovered later.