UNIT 2

MATHEMATICS LANGUAGE
AND SYMBOLS
CONTENT
• The Language of
Mathematics
• Expressions vs Sentences
• Unary and Binary
Operations
• Some Fundamentals of
Logic
HOW WILL YOU COMMUNICATE
WITH A SEATMATE IN THE BUS WHO
SPEAKS DIFFERENT LANGUAGE?
WHAT IS LANGUAGE?
Language is a systematic way of communication with other
people use of sounds or conventions symbols

IMPORTANCE :
Language was invented to communicate ideas to others

The language of mathematics was designed:

- Numbers
- Sets
- Functions
- Perform Operations (add, subt, multi, division)
ENGLISH VS. MATHEMATICS
ENGLISH VS. MATHEMATICS
CHARACTERISTICS OF

MATHEMATICAL LANGUAGE
Mathematical language is precise which means
it is able to make very fine distinctions or
definitions among a set of mathematical
symbols.

It is concise because a mathematician can

express otherwise long expositions or sentences
briefly using the language of mathematics

The mathematical language is powerful, that is,

one can express complex thoughts with relative
east.
CHARACTERISTICS OF

MATHEMATICAL LANGUAGE

“The sum of any two real numbers is also a real number”

In mathematical notation, this declarative sentence can be written as:

Mathematics is a symbolic language,

SYMBOLS COMMONLY USED IN MATHEMATICS:
• The ten digits (0, 1, ..... 9)
• Operations (+. -, X, division)
• Sets
• Variables: a, b, c, x, and y
• Special Symbols
• Logic Symbol
• Set Notations: N, W, Z, Q, R, C
CHARACTERISTICS OF

MATHEMATICAL LANGUAGE
CHARACTERISTICS OF

MATHEMATICAL LANGUAGE
CHARACTERISTICS OF

MATHEMATICAL LANGUAGE
Mathematical language can describe a
subset of the real world using only the
symbol above. Problem in physics like
freely falling bodies, speed, and
acceleration; quantities like chemical
content of vegetable: the use of
mathematical model in biological
disease modelling: and the formulas
employed in the social sciences can all
be expressed using mathematical
sentences or formulas.
SOME IMPORTANT SETS ARE THE FOLLOWING:

• N - reprsents as Natural Numbers = [1, 2, 3....] set of

natural numbers.
• W - Whole Number = [0, 1, 2, 3....] set of whole numbers
• Z - Integers [-3, -2, -1, 0, 1] set of integers
• Q - Rational Numbers = the set of rational numbers
• Q' - set of irrational numbers
• R - Real Number
• C - Complex Numbers
DIFFICULTIES:

• The word "is" could mean equality, inequality, or

member in a set.
• Different use of number (cardinal, ordinal, nominal,
ratio)
• Mathematical objects may be represented in many
ways such as sets and functions.
• The words "and" and "or" mean differently in
mathematics from its English use.
SOME ENGLISH PHRASES KEYWORDS
SOME ENGLISH PHRASES KEYWORDS
TRANSLATE EACH INTO ALGEBRAIC EXPRESSIONS. LET X REPRESENT A CERTAIN NUMBER
TRANSLATE EACH INTO ALGEBRAIC EXPRESSIONS. LET X REPRESENT A CERTAIN NUMBER
EXAMPLE :

“For any real number x, its square is greater than or equal to 0”

EXAMPLE :

“For any real number x, its square is greater than or equal to 0”

EXAMPLE :
“Ten if the square root of 100”
EXERCISE :
EXERCISE :
Write as mathematical sentences:

• Ten is greater than 9

• Ten is an even number

• Ten is a multiple of 5
REAL NUMBER SYSTEM (REVIEW IN ALGEBRA):
MATHEMATICAL VS. SENTENCES

The mathematical analogue of a ‘noun’ will be called an expression. Thus, an

expression is a name given to a mathematical object of interest. Whereas in

English we need to talk about people, places, and things, we’ll see that

mathematics has much different ‘objects of interest’.

The mathematical analogue of a ‘sentence’ will also be called a sentence. A

mathematical sentence, just as an English sentence, must state a complete

thought. The table below summarizes the apology.

MATHEMATICAL VS. SENTENCES
UNARY AND BINARY OPERATIONS

UNARY OPERATION
⚬ is an operation with only one operand
⚬ Common notations are prefix notation (e.g. +, −, ¬), postfix notation
(e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and
superscripts (e.g. transpose A^T). Other notations exist as well. For
example, in the case of the square root, a horizontal bar extending the
square root sign over the argument can indicate the extent of the
argument
UNARY AND BINARY OPERATIONS
UNARY OPERATION
⚬ is an operation with only one operand
⚬ Common notations are prefix notation (e.g. +, −, ¬), postfix notation
(e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and
superscripts (e.g. transpose A^T). Other notations exist as well. For
example, in the case of the square root, a horizontal bar extending the
square root sign over the argument can indicate the extent of the
argument
UNARY AND BINARY OPERATIONS
BINARY OPERATION
• on a set is a calculation that combines two elements of the set (called
operands) to produce another element of the set.
PROPERTIES OF BINARY OPERATIONS (ADDITION AND MULTIPLICATION)

COMMUTATIVITY OF BINARY OPERATIONS

• changing the order of the operands does not change the result.

COMMUTATIVITY OF BINARY OPERATIONS

• Within an expression containing two or more occurrences in a row of the
same associative operator, the order in which the operations are
performed does not matter as long as the sequence of the operands is
not changed
PROPERTIES OF BINARY OPERATIONS (ADDITION AND MULTIPLICATION)

DISTRIBUTIVITY OF BINARY OPERATIONS

• To multiply a sum (or difference) by a factor, each summand (or
minuend and subtrahend) is multiplied by this factor and the resulting
products are added (or subtracted).

IDENTITY ELEMENTS OF BINARY OPERATIONS

• When an identity element is paired with any element via the operation, it
returns that element.
PROPERTIES OF BINARY OPERATIONS (ADDITION AND
MULTIPLICATION)

INVERSES OF BINARY OPERATIONS

a+(−a)=0=e+ and (−a)+a=0=e+
PROPERTIES OF BINARY OPERATIONS (ADDITION AND MULTIPLICATION)

Commutative : a+b = b+a

Associative : (a+b) + c = a + (b+c)

Identity : a+0 = a

Inverse : a + (-a) = 0
THANK YOU
END OF UNIT 2
SEATWORK