**ENROLLED STUDENTS ONLY **

**MATH EXAM**

**INSTRUCTIONS EXAMS **

NOTE:

Selected statements shall be answered True or False. Please note that the statements making up those examinations may or may not be related to any summary that you have submitted for this subject.

Students sending answers for this Math examination through Email please uses the following format: M.01(T) or M.01(F).

MATH EXAM

STUDENTS SELECT 20 QUESTIONS FROM THE LIST

It is required that from the 20 questions selected from each exam, the student must correctly answer a minimum of 12 to approve the subject. If less than 12 correctly answered, the student must select the same number of questions failed from the list to complete 12. Example: you correctly answered 10, therefore you need 2 more correct answers to approve the subject. You may continue your selection as many times needed until approval.

**M.01. Mathematics** (from Greek μάθημα, knowledge, study, and learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of **mathematics**.

**M.02**. Aristotle defined mathematics as “the science of quantity”, and this definition prevailed until the 18th century.

**M.03.** Today, no consensus on the definition of mathematics prevails; even among professionals there is not even consensus on whether mathematics is an art or a science.

**M.04**.Practical mathematics has been a human activity for as far back as written records exists.

**M.05.** Galileo Galilei (1564-1642) said, “The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.”

**M.07**. Carl Friedrich Gauss (1777- 1855) known as the prince of mathematicians referred to mathematics as “the Queen of the Sciences”. More recently, Marcus du Sautody has called mathematics “the Queen of Science … the main driving force behind scientific discovery”.

**M.08**. Albert Einstein (1879-1955) 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.

**M.09**. Three leading types of definition of mathematics are called **logicist**, **intuitionist**, and **formalist,** each reflecting a different philosophical school of thought.

**M.10.** A **logicist **definition of mathematics is Russell’s “All Mathematics is Symbolic Logic” (1903).

**M.11. Formalist **definitions identify mathematics with its symbols and the rules for operating on them. A formal system is a set of symbols, or *tokens*, and some*rules* telling how the tokens may be combined into *formulas*.

**M.12. Applied mathematics** is a branch of mathematics that deal with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge.

**M.13. Abstraction** is the quality of dealing with ideas rather than events. The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals, was probably that of numbers.

**M.14.** A **number** is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2.3, and so forth.

**M.15. Algebra** (from Arabic *“al-jab”* meaning “reunion of broken parts) is one of the broad parts of mathematics, together with number theory, geometry and analysis. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra.

**M.16.** Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

**M.17. Physics** (from Ancient Greek: φυσική (ἐπιστήμη) phusikḗ (epistḗmē) “knowledge of nature”, from φύσις “nature”) is the natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force.

**M.18. Logicism** is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic.

**M.19. Geometry** (from the Ancient Greek γεωμετρία; geo “earth”, *-matron*” measurement”) is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

**M.20**. In geometry, a **hypotenuse** (alternate spelling: hypotenuse) is the longest side of a right-angled triangle, the side opposite of the right angle.

**M.21**. The **adjacent leg** is the other side that is adjacent to angle *A*.

**M.22. Euclidean geometry** is a mathematical system attributed to Euclid the Alexandrian Greek mathematician.

**M.23.** The **opposite side** is the side that is opposite to angle *A*. The terms **perpendicular **and **base** are sometimes used for the opposite and adjacent sides respectively.

**M.24. Trigonometry** (from Greek trigōnon, “triangle” and matron, “measure”) is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

**M.25. Statistics** is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying **statistics** to, e.g., a scientific, industrial, or societal problem, it is conventional to begin with a **statistical** population or **statistical** model process to be studied.

**M.26. Calculus** is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

**M.27. Axiom **is a statement or proposition that is regarded as being established, accepted, or self-evidently true.

**M.28**. As used in modern logic an **axiom **is simply a premise or starting point for reasoning. Thus, the axiom can be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.

**M.29**. As used in mathematics, the term *axiom* is used in two related but distinguishable senses: **“logical axioms” **and **“non-logical axioms”.**

**M.30**. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms).

**M.31**. In mathematics, a **theorem** is a statement that has been proven on the basis of previously established statements, such as other **theorem **and generally accepted statements, such as axioms. A **theorem** is a logical consequence of the axioms.

**M.32**. In mathematics, the **Pythagorean Theorem**, also known as **Pythagoras’s theorem**, is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

**M.33**. The **coordinates** are often chosen such that one of the numbers represents vertical position, and two or three of the numbers represent horizontal position. A common choice of **coordinates** is latitude, longitude and elevation.

**M.34**. The introduction of coordinates by Rene Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations.

**M.35**. It is common to see universities divided into sections that include a division of *Science and Mathematics*, indicating that the fields are seen as being allied but that they do not coincide.

**M.36**. Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (*A* and *B*) implies *A*), while non-logical axioms (e.g., *a* + *b* = *b* + *a*) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic).

M.37. In Euclid’s time, there was no clear distinction between physical and geometrical space. Since the discovery of non-Euclidean geometry in the 19^{TH} century, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space.

**M.38**. Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity.

**M.39**. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory.

**M.40**. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.

**M.41**. Albert Einstein, in his theory of **special relativity**, determined that the laws of physics are the same for all non-accelerating observers, and he showed that the speed of light within a vacuum is the same no matter the speed at which an observer travels.

**M.42.** In physics, **string theory** is a **theoretical** framework in which the point-like particles of particle physics are replaced by one-dimensional objects called **strings**. It describes how these **strings** propagate through space and interact with each other.

**M.43. Integers** are the set of whole numbers and their opposites. Whole numbers greater than zero are called positive **integers**. Whole numbers less than zero are called negative **integers**. The **integer **zero is neither positive nor negative, and has no sign.

**M.44**. Set Theory is the branch of mathematics that deals with the formal properties of sets as units (without regard to the nature of their individual constituents) and the expression of other branches of mathematics in terms of sets.

**M.45**. In mathematics, physics, and engineering, a Euclidean **vector** (sometimes called a geometric or spatial **vector**, a geometric object that has magnitude (or length) and direction) can be added to other **vectors** according to **vector** algebra.