Q.Compute $z$, $z^2$, $z^3$, $z^{-1}$ for the following $z$: $1 - i$, $3 + i$, $4 + 6i$, $\dfrac{9 + 2i}{2 + 9i}$, $1 + \pi i$, $\sqrt{3} + \sqrt{6}\,i$.

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