Mathematics as a Language

1. Chapter 2
MATHEMATICAL
LANGUAGE
AND
SYMBOLS

2. Imagine the following scenario.
You traveled through deep space to visit a planet called Sipnarys where everyone is a
genius Mathematician, the Sypnayans. You entered a coffee shop and you noticed two
Sypnayans talking. Here is a part of their conversation:
Sypnayan 1: Hagoudu estei freiou deimu
Sypnayan 2: Eyiedu estoureich salou
Sypnayan 1: Hyetie meich karhou shou leiou

3. The Language of Ordinary Speech (English Language) vs Mathematics
The English Language Mathematics
· Uses words · Uses numbers, operations,
sets, matrices, etc. in symbol
· A group of words conveying a
complete thought is an English
Sentence.
· A group of expressions
conveying a complete thought is
a Mathematical Sentence.
Equations are classic examples.
· A group of English sentences
comprises a paragraph.
· A group of equations is called a
system of equations.

4. The Language of Ordinary Speech (English Language) vs Mathematics
The English Language Mathematics
· Classifies which are nouns, verbs,
pronouns, adjectives, etc.
§ Nouns (Names, Places, Events)
§ Verbs (action words like listen)
§ Pronouns (he, she, it)
§ Adjectives (intelligent, hot)
§ Conjunctions (and, or, if, then)
· Do not exactly use the words "noun",
"verb", or "pronoun" but we can picture
likeness such as
§ Nouns (fixed things like numbers)
§ Verbs (is equal to, is greater than, etc.)
§ Pronouns (could be variables such as x
and y)
§ Adjective (of being prime, even, etc.)
§ Conjunctions (operations such as +, −, ×,
÷)

5. Characteristics of Mathematics as a Form of Language
Precise Concise Powerful
Mathematics is
precise by
making very
fine distinctions
among
mathematical
objects.
Mathematics is
concise because it
makes use of
symbols to convey
ideas, and that
what could be said
in thousands of
words may be
conveyed with few
Mathematics is
powerful
because it
expresses ideas
in ways that
allow the
solution of even
a complex

6. The Grammar of Mathematics
It is the structural rules governing the use of symbols representing
mathematical objects like expressions, variables and mathematical statements,
numbers, operations, sets, relations and functions.

7. Operations
Unary Operations are used on single mathematical objects.
Taking the additive inverse of a number and squaring it are
examples of a unary operation.
Examples:
Roots, Power/Exponents,
Binary Operations are used between two objects. The four
fundamental operations of mathematics are binary operations.
Examples:
Addition, Subtraction, Multiplication, Division,

8. Variables
A variable can be thought of as a mathematical “John Doe”
because it can be used as a placeholder or a symbol of something
that has one or more values.
Conventionally, we use letters as variables.
Example:
What number, when doubled and added to 1, is greater than
10?
Mathematically, 2x + 1 > 10. 4

9. Expressions
Expressions are mathematical ideas formed by
combining numbers and variables using the different
operations of mathematics.
Expressions are in their simpler forms if they involve
fewer symbols and operations.
Examples:
5, 2+x, x, 10x -2

10. Mathematical Statements
A mathematical statement is the analogue for an
English sentence. It should state a complete thought.
Examples:
x = 2
2x-y < 10
10-a > z

11. Practice Exercises
Symbolize the following statements.
1. The square of a number is always nonnegative.
A. x^2 > 1 B. x^2 ≥ 0
C. 1 < x^2 D. x^2 = 0

12. Practice Exercises
Symbolize the following statements.
2. The sum of two numbers is greater than their product.
A. ab > a + b B. a + b ≥ ab
C. ab ≥ a + b D. a + b > ab

13. Practice Exercises
Symbolize the following statements.
3. The quotient of two numbers less 3 is equal to 6.
A. a/b − 3 = 6 B. a (1/b) − 3 =6
C. a + b = 6 D.ab − 3 = 6

14. Practice Exercises
Symbolize the following statements.
4. A number subtracted from its cube is 9.
A. y – x^3 = 9 B. x^3 − y = 9
C. y – y^3 = 9 D. y^3 − y = 9

15. Practice Exercises
Symbolize the following statements.
5. Maria is 4 times younger than his brother (x). Which
gives the age of Maria (y)?
A. y = x B. y = 4x
C. x = 4y D. y = x^4

16. Types of Mathematical Statements
Universal Statements are those that hold true for all elements
of a set. In other words, these statements attribute a property to
all elements in a particular universe of discourse. They explicitly or
implicitly use universal quantifiers such as "all", "every" and
"each."
Example 3
1. The square of a real number is nonnegative.

17. Existential Statements attribute a property to at least
one object or entity, but not all, in a particular universe
of discourse.
Example
The following are examples of existential statements.
1. There exists a real number x such that 2x + 5 = 10.
2. There exist a prime number that is even.

18. Conditional Statements says that if one thing is true
then some other things are also true. It is a statement
that may be written in the form “If P then Q,” where P
and Q are simple propositions. Also, P is called the
hypothesis and Q is called the conclusion.
Example 3.1.4
The following are examples of conditional statements.
1. A polygon is a pentagon, if it has five sides.
2. If a rectangle is a square, then the adjacent sides are

19. Universal Conditional Statements are statements that
are both universal and conditional.
Example 3.1.5
The following are examples of universal conditional
statements.
1. All real numbers are rational if they can be
expressed as a ratio of two integers, where the
denominator is not zero.
2. For all numbers, if they are divisible by 6, then they
are also divisible by 2 and 3.

20. Exercise 2.1
Mathematics as a Form of Language
A. Identify whether the following statements are true or false.
1. Mathematics is a form of language.
2. 3x^2 − 4x + 1 is a correct mathematical sentence.
3. x+ x^2 = 2 is a correct mathematical sentence.
4. Variables are used to fancy mathematical ideas.
5. Variables are used when the value of something is unknown.

21. Exercise 2.1
Mathematics as a Form of Language
B. Translate the following sentences mathematically.
1. A number less its cube is zero.
2. The square of the sum of two numbers is 3 less than their product.
3. The difference of the squares of two numbers is greater than the
square of their difference.
4. The square root of the sum of three numbers is 4 more than their
product.
5. The cube root of the cube root of the square of a number is 1.

22. Exercise 2.1
Mathematics as a Form of Language
C. Identify the type of the following statements.
1. There exists a number that is both even and prime.
2. For all positive numbers, if it is less than one, then its cube is also
less than 1.
3. The square of a positive number is always positive.
4. For every real number, there is a corresponding multiplicative
inverse.
5. There exists a whole numbers less than every natural number.