Number System

What is the Number System?

The number system is the foundation of arithmetic and quantitative aptitude. It covers different types of numbers, how they behave, and how they interact. A good grasp of this topic builds a solid base for mastering topics like divisibility, HCF & LCM, and arithmetic operations.

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Types of Numbers

1. Natural Numbers

• Positive integers starting from 1
  Example: 1, 2, 3, 4, …

2. Whole Numbers

• All natural numbers including 0
  Example: 0, 1, 2, 3, …

3. Integers

• Positive and negative whole numbers, including 0
  Example: …, −3, −2, −1, 0, 1, 2, …

4. Rational Numbers

• Can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$
  Examples: $\frac{2}{3}$, 0.75, −4

5. Irrational Numbers

• Cannot be expressed in the form $\frac{p}{q}$
• Non-terminating, non-repeating decimals
  Examples: $\sqrt{2}$, $\pi$, $\phi$

6. Real Numbers

• Union of rational and irrational numbers
• Represent all points on the number line

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Comparison Table

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Key Concepts

Even and Odd Numbers

• Even: Divisible by 2 → 0, 2, 4, …
• Odd: Not divisible by 2 → 1, 3, 5, …

Prime Numbers

• Greater than 1
• Exactly two distinct positive divisors: 1 and the number itself
  Examples: 2, 3, 5, 7, 11
  Note: 2 is the only even prime number

Composite Numbers

• Natural numbers greater than 1 that are not prime
  Examples: 4, 6, 8, 9, 10

Co-prime Numbers

• Two numbers whose HCF is 1
  Example: 14 and 25 (HCF = 1)

Perfect Numbers

• Sum of proper divisors (excluding the number itself) equals the number
  Example:

  • Divisors of 28: 1, 2, 4, 7, 14
  • Sum = 1 + 2 + 4 + 7 + 14 = 28 → Perfect Number

Face Value vs. Place Value

• Face Value: The digit itself
• Place Value: Digit × positional weight

Example (in 4826):

• Face value of 8 = 8
• Place value of 8 = $8 \times 100 = 800$

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Prime Number Check (Root $n$ Method)

To check if a number $n$ is prime:

1. Find $\sqrt{n}$
2. Check for divisibility by all primes ≤ $\sqrt{n}$

If it is not divisible by any of those, it's a prime.

Example:
Is 97 a prime number?

• $\sqrt{97} \approx 9.8$

• Check divisibility by 2, 3, 5, and 7

• Not divisible by any of them

• Hence, 97 is a prime

This method is highly efficient for numbers < 1000.

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Digital Sum and Digital Root

Digital Sum

• Sum of digits of a number
  Example: 537 → 5 + 3 + 7 = 15

Digital Root

• Repeated sum until a single digit remains
  Example: 537 → 15 → 1 + 5 = 6

Uses:

• Quick divisibility checks for 3 and 9
• Option elimination in MCQs

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Important Formulas and Techniques

1. Number of Digits in a Number

$$\text{Number of digits} = \lfloor \log_{10} N \rfloor + 1$$

Example:
$\log_{10}(2) \approx 0.3010$
To find digits in $2^{30}$:

$$\lfloor 30 \times 0.3010 \rfloor + 1 = \lfloor 9.03 \rfloor + 1 = 10$$

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2. General Form of Division

If a number $N$ is divided by $d$ and gives remainder $r$:

$$N = dq + r$$

Where:

• $d$: divisor
• $q$: quotient
• $r$: remainder

Used in modular arithmetic, remainder theorems, and divisibility problems.

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Common Mistakes to Avoid

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Sample Practice Questions

Q1. How many digits are in $2^{30}$?
Solution:

$$\log_{10}(2) \approx 0.3010 \Rightarrow 30 \times 0.3010 = 9.03 \\ \Rightarrow \lfloor 9.03 \rfloor + 1 = 10 \text{ digits}$$

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Q2. Is 91 a prime number?
Solution:
$91 = 7 \times 13 \Rightarrow$ Not prime

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Q3. Is 49 a perfect number?
Divisors of 49: 1, 7
Sum: 1 + 7 = 8 ≠ 49 → Not perfect