Imaginary unit i = √(−1), i² = −1, powers of i cycle: Foundations

This is the foundations floor for Complex Numbers and specifically for the parent note on the imaginary unit $i$ . We will not assume you already know what , exponents, "remainder", or "the plane" mean. We build each one from zero, in an order where every idea leans only on the ones before it.

1. The number line — where "real numbers" live

Look at Figure 1. The line stretches forever both ways. Numbers to the right of $0$ are positive; numbers to the left are negative. Every real number is one dot on this one line — there is nowhere else to go.

Figure — Imaginary unit i = √(−1), i² = −1, powers of i cycle

2. The symbols $+$ , $-$ , and the meaning of a sign

Why do we fuss over signs? Because the entire reason $i$ is needed comes from one fact about signs and multiplication, which is our next tool.

3. Multiplication and the sign rule

The one fact we need next is the sign rule for multiplying:

4. Squaring, and why nothing real squares to a negative

Now combine squaring with the sign rule. Whatever real number $a$ you pick:

If $a$ is positive: $a^{2} = (+) \times (+) = +$ .
If $a$ is negative: $a^{2} = (-) \times (-) = +$ .
If $a = 0$ : $a^{2} = 0$ .

5. The square root symbol — the "undo" of squaring

6. Whole numbers, division, and the remainder (the $mod$ symbol)

The four-cycle in the parent note uses a "remainder". This is the symbol that runs the whole cycle, so we build it before we lean on it.

7. Letters as stand-ins: variables $a$ , $b$ , $n$ , $k$ , $x$

Now that we have exponents (Section 4) and the remainder symbol (Section 6), we can read the parent note's headline formula. But formulas are written with letters, so we pin those down first.

Why so many letters? Because the parent's headline formula $i^{n} = i^{n mod 4}$ is a statement about every exponent $n$ at once — using the remainder $n mod 4$ from Section 6 — and you cannot say "for every $n$ " with a single fixed number, only with a variable.

What is a real number, in one sentence? ::: Any number that sits as a single point on the endless number line.
What does the sign of a number tell you? ::: Which side of $0$ it is on — right is positive, left is negative.
What is the picture of multiplication that works for fractions and irrationals too? ::: Scaling — stretching a length by the factor $b$ , not just repeated adding.
Why does a negative times a negative give a positive? ::: Multiplying by a negative flips you across $0$ ; two flips return you to the positive side.
What does the exponent in $a^{n}$ count? ::: How many copies of the base $a$ are multiplied together.
Why can no real number square to $- 1$ ? ::: By the sign rule any real squared is $\geq 0$ , so it can never be negative.
What question does $x$ ask, and which of the two answers does it pick? ::: "Which number squared gives $x$ ?" — and by the principal-root convention it picks the non-negative one, so $9 = 3$ , not $- 3$ .
What does $n mod 4$ mean? ::: The remainder left after removing as many whole $4$ s as possible from $n$ .
Compute $27 mod 4$ . ::: $3$ , since $27 = 4 \times 6 + 3$ .
What is a degree, and how many make one full turn? ::: A degree is one of $360$ equal slices of a full turn; a quarter turn is $9 0^{\circ}$ .
Why is the plane needed for $i$ ? ::: A single line is full of reals; a second, vertical direction gives $i$ a home off the line.
How many $9 0^{\circ}$ turns make a full circle? ::: Four, because $4 \times 9 0^{\circ} = 36 0^{\circ}$ .

Connections

Complex Numbers — the parent framework $a + bi$ these foundations unlock.
Argand Diagram — the plane from Section 8, made precise.
Modulus and Argument — measuring distance and angle on that plane.
Roots of Unity — where the four clock marks $1, i, - 1, - i$ reappear.
Euler's Formula — turns the $9 0^{\circ}$ rotation into an angle $π /2$ .
Solving Quadratic Equations — the source of the equation $x^{2} = - 1$ that forces $i$ .
Hinglish version of the parent →

Foundations — Imaginary unit i = √(−1), i² = −1, powers of i cycle