PPT: Number System | Quantitative Aptitude (Quant) - CAT PDF Download

 Page 1


Number System
Page 2


Number System
Introduction to Numbers
Definition: Numbers are mathematical entities 
used for counting, measuring, or labeling.
Importance: Foundation of arithmetic and key 
for Quantitative Section.
Page 3


Number System
Introduction to Numbers
Definition: Numbers are mathematical entities 
used for counting, measuring, or labeling.
Importance: Foundation of arithmetic and key 
for Quantitative Section.
Types of Numbers
1
Natural Numbers (N)
{1, 2, 3, ...} - Counting numbers.
2
Whole Numbers (W)
{0, 1, 2, ...} - Natural numbers + zero.
3
Integers (Z)
{..., -2, -1, 0, 1, 2, ...} - Positive, negative, and zero.
4
Rational Numbers
p/q where q b 0 (e.g., 1/2, -3/4).
5
Irrational Numbers
Cannot be expressed as p/q (e.g., :2, Ã).
6
Real Numbers
All rational and irrational numbers.
7
Imaginary Numbers
Involve i where i² = -1 (e.g., 2i, 1 + 4i).
Example: Classify 5, -3/2, :3
5: Natural, Whole, Integer, Rational, Real
-3/2: Rational, Real
:3: Irrational, Real
Page 4


Number System
Introduction to Numbers
Definition: Numbers are mathematical entities 
used for counting, measuring, or labeling.
Importance: Foundation of arithmetic and key 
for Quantitative Section.
Types of Numbers
1
Natural Numbers (N)
{1, 2, 3, ...} - Counting numbers.
2
Whole Numbers (W)
{0, 1, 2, ...} - Natural numbers + zero.
3
Integers (Z)
{..., -2, -1, 0, 1, 2, ...} - Positive, negative, and zero.
4
Rational Numbers
p/q where q b 0 (e.g., 1/2, -3/4).
5
Irrational Numbers
Cannot be expressed as p/q (e.g., :2, Ã).
6
Real Numbers
All rational and irrational numbers.
7
Imaginary Numbers
Involve i where i² = -1 (e.g., 2i, 1 + 4i).
Example: Classify 5, -3/2, :3
5: Natural, Whole, Integer, Rational, Real
-3/2: Rational, Real
:3: Irrational, Real
PYQ
Q: If (a + b :n) is the positive square root of ( 29 2 12 :5), where a and b are integers, and n is a 
natural number, then the maximum possible value of (a + b + n ) is 
Ans:  18
Sol:  We are given that: 
Squaring both sides: 
Equating the rational and irrational parts:
- a^2 + b^2n = 29 (rational part) - 2ab:n =- 12:5 (irrational part)
From 2ab:n =- 12:5, comparing the terms under the square root gives n = 5, so: 
Now, using a^2 + b^2n = 29, we substitute n = 5:
a2 + 5b^2 = 29
We have two equations: 1. ab =- 6 
2. a^2 +5b^2 = 29
By trial and error or systematic solving, we find a = 3, b = -2, and n = 5.
Thus, a + b + n = 3 - 2 + 5 = 6.
Page 5


Number System
Introduction to Numbers
Definition: Numbers are mathematical entities 
used for counting, measuring, or labeling.
Importance: Foundation of arithmetic and key 
for Quantitative Section.
Types of Numbers
1
Natural Numbers (N)
{1, 2, 3, ...} - Counting numbers.
2
Whole Numbers (W)
{0, 1, 2, ...} - Natural numbers + zero.
3
Integers (Z)
{..., -2, -1, 0, 1, 2, ...} - Positive, negative, and zero.
4
Rational Numbers
p/q where q b 0 (e.g., 1/2, -3/4).
5
Irrational Numbers
Cannot be expressed as p/q (e.g., :2, Ã).
6
Real Numbers
All rational and irrational numbers.
7
Imaginary Numbers
Involve i where i² = -1 (e.g., 2i, 1 + 4i).
Example: Classify 5, -3/2, :3
5: Natural, Whole, Integer, Rational, Real
-3/2: Rational, Real
:3: Irrational, Real
PYQ
Q: If (a + b :n) is the positive square root of ( 29 2 12 :5), where a and b are integers, and n is a 
natural number, then the maximum possible value of (a + b + n ) is 
Ans:  18
Sol:  We are given that: 
Squaring both sides: 
Equating the rational and irrational parts:
- a^2 + b^2n = 29 (rational part) - 2ab:n =- 12:5 (irrational part)
From 2ab:n =- 12:5, comparing the terms under the square root gives n = 5, so: 
Now, using a^2 + b^2n = 29, we substitute n = 5:
a2 + 5b^2 = 29
We have two equations: 1. ab =- 6 
2. a^2 +5b^2 = 29
By trial and error or systematic solving, we find a = 3, b = -2, and n = 5.
Thus, a + b + n = 3 - 2 + 5 = 6.
Classification of Numbers
Even Numbers
Divisible by 2 (e.g., 2, 4, 0); Form: 2n.
Odd Numbers
Not divisible by 2 (e.g., 1, 3, -5); Form: 2n + 1.
Prime Numbers
Exactly 2 factors (1 and itself); e.g., 2, 3, 5. 2 is the only even prime.
Prime > 3: Form 6k ± 1 (e.g., 5 = 6·1 - 1).
Composite Numbers
More than 2 factors (e.g., 4, 6); Standard form: p ¡ _·p¢ _.
Perfect Numbers
Sum of proper factors = number (e.g., 6: 1+2+3=6).
Fractions:
Proper: Numerator < Denominator (e.g., 2/3).
Improper: Numerator > Denominator (e.g., 7/3).
Mixed: Whole + Proper (e.g., 2 1/3).
Decimals:
T erminating (e.g., 0.25).
Non-terminating: Recurring (e.g., 0.333&) or Non-recurring (e.g., :2).
Example: Is 25 prime?
Factors: 1, 5, 25 (more than 2) ³ Composite.