Q.Find the complex conjugates and modulus of the following complex numbers: $1 - i$, $10 + 4i$, $(3 + 5i)(4 + 6i)$, $\dfrac{2 + 7i}{5 + 4i}$.

What is a Prime Number?

Imagine you have a pile of identical coins. You want to arrange them into a neat rectangle — rows and columns, with no gaps and no leftover coins. For some numbers of coins, you can do this in more than one way. For others, you can only make a single row (or a single column). Those stubborn numbers that refuse to form any rectangle except a straight line are the prime numbers.

Take 6 coins. You can arrange them as 1 row of 6, 2 rows of 3, 3 rows of 2, or 6 rows of 1. That's four different rectangles. Now take 7 coins. You can only make 1 row of 7 or 7 rows of 1 — nothing else. 7 is prime.

So the core idea is simple: a prime number cannot be split into equal groups (other than groups of 1 or the number itself). It is "indivisible" in that sense.

The Precise Definition

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

A composite number is a natural number greater than 1 that has more than two positive divisors.

The number 1 is neither prime nor composite — it has only one divisor (itself), so it doesn't fit either category.

How to Check if a Number is Prime

To test if a number $n$ is prime, you check whether any number from 2 up to $\sqrt{n}$ divides $n$ evenly. If none do, $n$ is prime.

Why only up to $\sqrt{n}$? Because if $n = a \times b$ and both $a$ and $b$ are greater than $\sqrt{n}$, then $a \times b > n$. So at least one factor must be $\le \sqrt{n}$.

Example: Is 29 prime? $\sqrt{29} \approx 5.4$. Check divisibility by 2, 3, 5. None divide 29. So 29 is prime.

The First Few Primes

Notice that 2 is the only even prime. Every other even number is divisible by 2, so it has at least three divisors (1, 2, and itself).

Why Primes Matter

Primes are the building blocks of all numbers. Every integer greater than 1 can be written as a product of primes in exactly one way (ignoring order). This is the Fundamental Theorem of Arithmetic.

For example:

• $12 = 2 \times 2 \times 3$
• $30 = 2 \times 3 \times 5$
• $100 = 2 \times 2 \times 5 \times 5$

This uniqueness is why primes are so fundamental — they are like the atoms of the number system.

A Quick Summary

• A prime has exactly two divisors: 1 and itself.
• 1 is not prime.
• 2 is the only even prime.
• To test if $n$ is prime, check divisibility by primes up to $\sqrt{n}$.
• Every number can be broken down uniquely into prime factors.

That's the essence. Once you see numbers as either "prime" or "composite", you start noticing patterns — and that's where the real beauty of number theory begins.