Multiplying by i.

Step 1 of 6

The square of i

No real number has a negative square: $2^2 = 4$⁠ and $(-2)^2 = 4$⁠ too. Mathematicians added a new number, $i$⁠, defined so that $i^2 = -1$⁠, and found that it contradicts nothing. A complex number is $a + bi$⁠, with $a$⁠ and $b$⁠ real: $a$⁠ is the real part and $b$⁠ the imaginary part. Every such number is a point in the plane, $a$⁠ across and $b$⁠ up, like a vector $(a, b)$⁠. This is the complex plane.

On the board the point starts at $1$⁠, and each step multiplies it by a number $w$⁠. Here $w = i$⁠: Turn per step is $90^\circ$⁠ and Stretch per step is $1$⁠. Each point is raised above the board by its step number, so the products climb a spiral staircase. Predict where the point will be after two steps.

Step 2 of 6

The powers of i

Multiplying by $i$⁠ turns any number a quarter turn: $1 \cdot i = i$⁠, $i \cdot i = -1$⁠, $-1 \cdot i = -i$⁠, $-i \cdot i = 1$⁠. The powers of $i$⁠ repeat every four. Move Steps and follow the card, which shows the last product.

Step 3 of 6

Length and angle

Like a vector, a complex number can also be described by a length $r$⁠ and an angle $\varphi$⁠ from the real axis:

$w = r(\cos\varphi + i\sin\varphi)$$w =$$r(\cos\varphi + i\sin\varphi)$⁠

Here $w$⁠ turns by $30^\circ$⁠ and has length $1$⁠, and Steps is locked at $0$⁠: the board shows only the starting point, $1$⁠, and $w$⁠ on the card as a turn and a stretch. Write $w$⁠ in the form $a + bi$⁠.

Step 4 of 6

Roots of minus 1

If one multiplication by $w$⁠ turns by $\varphi$⁠, then $w^3$⁠ turns by $3\varphi$⁠. Find the Turn per step whose three steps bring $1$⁠ to $-1$⁠.

Step 5 of 6

Squaring

To multiply $(a + bi)(c + di)$⁠, open the brackets as with algebraic expressions, and remember that $i^2 = -1$⁠:

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$(a + bi)(c + di) =$$(ac - bd) + (ad + bc)i$⁠

Choose your answer, then check it on the board with Steps.

Step 6 of 6

The imaginary part

A number whose length is not $1$⁠ can also be written by its length and angle. The real part is how far right it is, and the imaginary part how far up.