04-Mathematical-Language-and-Symbols.pptx

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1. MATHEMATICAL LANGUAGE AND SYMBOLS

2. Mathematics relies on a specialized language and symbols to express concepts, equations, and relationships precisely and concisely. Here are some fundamental mathematical symbols and notations:

3. Numbers : Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} Natural numbers: {1, 2, 3, 4, ...} Real numbers: All numbers that can be represented on the number line, including integers, fractions, and irrational numbers like π (pi) and √2 (square root of 2).

4. Basic Arithmetic : Addition (+) Subtraction (-) Multiplication (× or *) Division (÷ or /) Equals (=)

5. Common Mathematical Operations : Exponentiation (^ or **): 2^3 = 2 × 2 × 2 = 8 Square root (√): √9 = 3 Absolute value (|x|): |−5| = 5 Factorial (!): 5! = 5 × 4 × 3 × 2 × 1 = 120

6. Basic Relations : Less than (<) Greater than (>) Less than or equal to (≤) Greater than or equal to (≥) Not equal to (≠)

7. Sets and Set Notation : Set: {1, 2, 3} Subset (⊆): A ⊆ B (A is a subset of B) Union (∪): A ∪ B (the union of sets A and B) Intersection (∩): A ∩ B (the intersection of sets A and B)

8. Algebraic Notation : Variables (usually represented by letters): x, y, z Algebraic expressions: 2x + 3y Equations: 2x + 3y = 10 Inequalities: x > 5

9. Calculus Notation : Derivative (d/dx or ∂/∂x): The rate of change of a function with respect to its variable. Integral (∫): Represents the accumulation of a function over an interval.

10. Geometry Symbols : π (pi): The mathematical constant representing the ratio of a circle's circumference to its diameter (approximately 3.14159). ∠ (angle): Used to denote angles in geometry. ∥ (parallel): Indicates that two lines are parallel. ⊥ (perpendicular): Indicates that two lines are perpendicular.

11. Greek Letters : α ( alpha), β ( beta), γ ( gamma), δ ( delta), θ ( theta), λ ( lambda), ω ( omega), etc. are often used to represent various quantities and variables in mathematics.

12. Summation and Product Notation : Σ (sigma): Represents a summation (adding up a series). Π (pi): Represents a product (multiplying a series).

13. Limit Notation : lim : Represents the limit of a function as a variable approaches a certain value . Complex Numbers : I : The imaginary unit, where i^2 = -1. Complex numbers: a + bi, where a and b are real numbers.

14. These are just some of the fundamental symbols and notations used in mathematics. As you delve deeper into various branches of mathematics, you'll encounter more specialized symbols and notations specific to those fields.

15. Expressions vs. Sentences

16. A sentence must contain a complete thought. In the English language, an ordinary sentence must contain a subject and a predicate. The subject contains a noun or a whole clause. “Manila” for example is a proper noun but is not in itself a sentence because it does not state a complete thought. Similarly, a mathematical sentence must state a complete thought. An expression is a name given to a mathematical expression but not a mathematical sentence.

17. Types of Mathematical Sentences A mathematical sentence is one in which a fact or complete idea expressed. Because a mathematical sentence states a fact, many of them can be judged to be “true” or “false”. Questions and phrases are not mathematical sentences since they cannot be judged to be true or false.

18. Examples: a. “An isosceles triangle has two congruent sides.” is a true mathematical sentence. b. “10 + 4 = 15” is a false mathematical sentence. c. “Did you get that one right?” is NOT a mathematical sentence – it is a question. d. “All triangles” is NOT a mathematical sentence – it is a phrase.

19. There are two types of mathematical sentences: Open Sentence An open sentence is a sentence which contains a variable. It can be either true or false depending on what values are used.

20. Examples: 1. A triangle has n sides. 2. z is a positive number. 3. 3y = 4x + 2 4. a + b = c + d

21. Closed Sentence A closed sentence is a sentence which can be judged to be always true or always false and has no variables. Examples: 1. A square has four corners. 2. 6 is less than 5. 3. −3 is a negative number. 4. 3 + 5 = 8 5. 9 is an even number

22. INSTRUCTION: Tell whether if each of the following sentences is an open sentence or a closed sentence. Write OS if a sentence is open and CS if it is closed. If CS , determine if it is true or false .

23. _____________________1. Nine is an even number . _____________________ 2. 4x – 2 = 5 _____________________ 3. Fourteen is an even number. _ ____________________4 . 2 + 5 = 2x _____________________ 5. 12 > 23 _____________________ 6. n is an odd number _____________________ 7. 2n < 5 _____________________ 8. – 1 is an integer. _____________________ 9. 2a + 3b = c + d ____________________ 10. 0 is not an integer.