Number System

QUIZ

Number Basics

Numbers form the foundation of all mathematical reasoning, and they can be organized into well-defined types that together create a unified numerical system. The major categories are:

  • Natural Numbers: \(\mathbb{N} = \{1,2,3,\dots\}\)
  • Whole Numbers: \(\{0,1,2,3,\dots\}\)
  • Integers: \(\mathbb{Z} = \{\dots,-3,-2,-1,0,1,2,3,\dots\}\)
  • Rational Numbers: \(\mathbb{Q} = \left\{\frac{p}{q} : p,q \in \mathbb{Z},\, q \neq 0\right\}\)
  • Irrational Numbers: numbers not expressible as any fraction, e.g., \(\sqrt{2},\,\pi,\,e\)
  • Real Numbers: \(\mathbb{R} = \mathbb{Q} \cup (\text{irrationals})\)
  • Complex Numbers: \(\mathbb{C} = \{a + bi : a,b \in \mathbb{R}\}\)

All these categories work together to build the complete number system. Natural and whole numbers support counting, integers allow direction, rational and irrational numbers fill the number line, and real numbers create a continuous spectrum. Complex numbers extend this by capturing quantities involving the imaginary unit \(i=\sqrt{-1}\). Despite their differences, all types interact through consistent rules, forming the backbone of algebra and quantitative reasoning.

Cyclic Behaviour of Last Digits

Many number system problems rely on the fact that the last digit of powers repeats in a cycle.
For example, powers of 7 follow the pattern:

7 → 9 → 3 → 1 → (repeats)

Since 17 ends with 7, the powers of 17 have the same last-digit cycle.
Thus, questions like how many numbers in
{17¹, 17², …, 17³⁰⁰}
end in digit 3 become a matter of cycle counting, not calculation.

This idea makes huge powers manageable and predictable.

Comparing Large Exponential Numbers

Expressions such as
2⁵⁰, 3⁴⁰, 4³⁰, 5²⁰
cannot be computed directly. Instead, we convert them to comparable bases.

For example:
4³⁰ = (2²)³⁰ = 2⁶⁰

Now everything is expressed in terms of powers of 2, and comparison becomes simple.

Understanding such transformations allows large exponent problems to be solved logically instead of numerically.

Prime Factorization and Divisibility Structure

Prime factorization is central to divisibility, gcd–lcm, perfect squares, and factor-counting questions.
Every integer has a unique prime factor representation.

Knowing this structure helps determine:

  • Whether a number is divisible by another
  • How many factors it has
  • Whether it is a perfect square
  • The gcd or lcm of two numbers

Factorization is not mere arithmetic; it reveals the internal geometry of numbers.