Pre-Calculus Unit 1.pdf By Dr. Mamoona Anam

✅ What Are Real Numbers?
Real numbers are all the numbers that can be found on the number line. This includes both rational and irrational numbers.

📚 Types of Real Numbers:
1. Rational Numbers (can be written as a fraction
𝑝
𝑞
q
p
​
, where
𝑞
≠
0
q

=0):
Integers:
…
,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
…
…,−3,−2,−1,0,1,2,3,…

Whole Numbers:
0
,
1
,
2
,
3
,
…
0,1,2,3,…

Natural Numbers:
1
,
2
,
3
,
…
1,2,3,…

Terminating or repeating decimals:

0.5
=
1
2
,
0.333
…
=
1
3
0.5=
2
1
​
,0.333…=
3
1
​

2. Irrational Numbers (cannot be written as a fraction; non-repeating, non-terminating decimals):
Examples:

𝜋
,
2
,
3
,
𝑒
π,
2
​
,
3
​
,e
🔢 Examples of Real Numbers:
5
5 → Natural number

−
4
−4 → Integer

0.75
0.75 → Rational

2
2
​
→ Irrational

𝜋
≈
3.14159
π≈3.14159 → Irrational

📈 On the Number Line:
Every point on the number line represents a real number. Real numbers can be:

Positive or negative

Whole or fractional

Repeating, terminating, or non-repeating decimals

⚠️ Not Real Numbers:
Imaginary numbers (like
−
1
−1
​
) are not real

Example:
−
9
=
3
𝑖
−9
​
=3i → Not a real number

🧠 In Summary:
Real numbers = Rational numbers + Irrational numbers
They represent all the values that can be plotted on the number line.

1. PRE-CALCULUS UNIT 1
BY
DR.MAMOONA ANAM

2. WHAT ARE REAL NUMBERS?
REAL NUMBERSARE ALL THE NUMBERSTHAT CAN BE FOUND ON THE NUMBER LINE. THEY
INCLUDE BOTH RATIONAL AND IRRATIONAL NUMBERS.
•THESE ARE THE NUMBERS WE USE IN EVERYDAY LIFE —FOR COUNTING, MEASURING,
CALCULATING, ETC. REAL NUMBERS ARE USED IN ALL FIELDS: MATHEMATICS, SCIENCE,
ENGINEERING, FINANCE, AND MORE.

3. Types of Real Numbers
Real numbers can be divided into two main categories:
1. Rational Numbers (ℚ)
These are numbers that can be written as a fraction(a/b) where:
•aand bare integers
•b ≠ 0
Examples:
•Integers: -3, 0, 4 (because -3 = -3/1, 0 = 0/1, etc.)
•Fractions: 2/3, -5/7
•Terminating decimals: 0.75 = 3/4
•Repeating decimals: 0.333… = 1/3
If a decimal endsor repeats, it's a rational number.

4. 2. IRRATIONAL NUMBERS
THESE ARE NUMBERS THAT CANNOT BE WRITTEN AS A FRACTIONOF TWO INTEGERS. THEIR DECIMAL
EXPANSIONS:
•GO ON FOREVER
•DO NOT REPEAT
EXAMPLES:
•√2 = 1.4142135… (NON-REPEATING AND NON-ENDING)
•Π= 3.14159265… (NON-REPEATING AND NON-ENDING)
•E = 2.7182818…
•THESE NUMBERS NEVERBECOME A NEAT FRACTION OR PATTERN.

5. Hierarchy of Real Numbers
•REAL NUMBERS
•├──RATIONAL NUMBERS (FRACTIONS, INTEGERS, DECIMALS THAT END OR REPEAT)
•│├──INTEGERS (..., -3, -2, -1, 0, 1, 2, 3, ...)
•││├──WHOLE NUMBERS (0, 1, 2, 3, ...)
•│││└──NATURAL NUMBERS (1, 2, 3, ...)
•│
•└──IRRATIONAL NUMBERS (Π, √2, E, ETC.)

6. WHAT ARE NOT REAL NUMBERS?
SOME NUMBERS DON’T FALL ON THE REAL NUMBER LINE:
•IMAGINARY NUMBERS: LIKE √(-1) = I
•COMPLEX NUMBERS: LIKE 2 + 3I
•THESE ARE NOT PART OF THE REAL NUMBER SYSTEM.

7. WHERE ARE REAL NUMBERS USED?
•EVERYWHERE!
•TEMPERATURE: -10°C, 37.5°C →REAL NUMBERS
•MONEY: $10.99, -$2.50 →REAL NUMBERS
•GEOMETRY: LENGTHS LIKE √2 CM
•SCIENCE: PH VALUES, MEASUREMENTS, TIME

8. SUMMARY
Type Example Notes
Natural Numbers 1, 2, 3 Counting numbers
Whole Numbers 0, 1, 2 Natural + 0
Integers -3, 0, 4 No fractions
Rational Numbers 2/3, -5, 0.75 Can be written as a fraction
Irrational Numbers π, √2 Cannot be written as a fraction
Real Numbers All above All numbers on the number line

9. OBJECTIVES OF THE REAL NUMBER SYSTEM
•HERE ARE THE MAIN GOALS OR OBJECTIVESOF THE REAL NUMBER SYSTEM:

10. 1. TO REPRESENT QUANTITIES ACCURATELY
•REAL NUMBERS ARE USED TO EXPRESS ALL POSSIBLE QUANTITIES, INCLUDING:
•HEIGHTS (E.G., 1.75 METERS)
•TEMPERATURES (E.G., -5°C)
•MONEY (E.G., $9.99)
•DISTANCES, SPEED, TIME, ETC.
•THIS SYSTEM ALLOWS FOR BOTH EXACT AND APPROXIMATE VALUES(LIKE ΠOR √2).

11. 2. TO ENABLE ARITHMETIC OPERATIONS
•REAL NUMBERS SUPPORT:
•ADDITION, SUBTRACTION
•MULTIPLICATION, DIVISION
•THESE OPERATIONS FOLLOW CONSISTENT RULES (CALLED FIELD PROPERTIES), MAKING MATH
RELIABLE AND SYSTEMATIC.

12. 3. TO FORM THE FOUNDATION OF ALGEBRA
AND CALCULUS
•REAL NUMBERS ARE ESSENTIALIN:
•ALGEBRA (SOLVING EQUATIONS, INEQUALITIES)
•GEOMETRY (LENGTHS, ANGLES)
•TRIGONOMETRY
•CALCULUS(LIMITS, DERIVATIVES, INTEGRALS)
•WITHOUT THE REAL NUMBER SYSTEM, THESE BRANCHES OF MATH WOULDN'T WORK.

13. 4. TO DISTINGUISH BETWEEN RATIONAL AND
IRRATIONAL NUMBERS
•HELPS IN CLASSIFYINGNUMBERS CLEARLY.
•RATIONAL NUMBERS ALLOW EXACT SOLUTIONS, WHILE IRRATIONAL NUMBERSHELP IN
DESCRIBING THINGS LIKE CIRCLE AREA (ΠR²) OR DIAGONAL OF A SQUARE (√2).

14. 5. TO MODEL REAL-WORLD PROBLEMS
•THE REAL NUMBER SYSTEM IS USED TO CREATE MATHEMATICAL MODELSFOR:
•PHYSICS
•ENGINEERING
•ECONOMICS
•BIOLOGY
•STATISTICS AND MORE
•FOR EXAMPLE, REAL NUMBERS ARE USED TO PREDICT POPULATION GROWTH,CALCULATE
INTEREST RATES, OR ANALYZE WAVES IN PHYSICS.

15. 6. TO ENSURE CONTINUITY ON THE NUMBER
LINE
THE REAL NUMBER SYSTEM IS CONTINUOUS, MEANING:
•THERE’S NO GAPBETWEEN NUMBERS.
•YOU CAN ALWAYS FIND ANOTHER NUMBER BETWEEN ANY TWO REAL NUMBERS.
•THIS IS IMPORTANT FOR PRECISE MEASUREMENTS AND SMOOTH GRAPHS.

16. 7. TO DEVELOP LOGICAL AND ABSTRACT
THINKING
•WORKING WITH REAL NUMBERS STRENGTHENS:
•PROBLEM-SOLVING SKILLS
•LOGICAL REASONING
•ABSTRACT THINKING
•THESE ARE ESSENTIAL IN BOTH ACADEMICS AND PRACTICAL LIFE

17. SUMMARY OBJECTIVES OF THE REAL NUMBERS
Objective Purpose
Represent quantities Accurately describe real-world values
Perform arithmetic Enable consistent calculations
Foundation for advanced math Essential for algebra, geometry, calculus
Classify numbers Understand rational vs irrational
Model real-world problems Used in science, finance, engineering
Ensure continuity No gaps on the number line
Develop logical thinking Boosts reasoning and problem-solving abilities