Monday
Monday: The Arc That Never Lands
Dawn light fills the old greenhouse behind the community center. Comet sets a sprinkler on the long bench and turns the tap.
The water lifts in an arc, hangs, and falls back onto the bench. Above it, a planter swings from a hook, dry.
"It never reaches the planter," Comet says. "What can we make that fixes this?"
Wren is already sketching. "First, a rule. Our model from yesterday's tape readings is h = -x² + 4x. What do you notice?"
"The planter base is at height 5," Comet says. "So set -x² + 4x equal to 5."
Nova hovers over the chalkboard, her light steady. "Would you like a hint? Try the quadratic formula and watch the root."
"Sixteen minus twenty," Wren says. "The square root of negative four."
"So the arc never lands there," Comet says. "But Nova, what number squares to negative four?"
A new number
No real number squares to a negative. Squares of positives are positive, and squares of negatives are positive too.
So mathematicians defined a new number, i, with one rule: i² = -1. It is called the imaginary unit.
Then √(-4) = √4 × √(-1) = 2i. And (2i)² = 4 × i² = 4 × (-1) = -4. It works.
A complex number has the form a + bi. For 3 + 2i, the real part is 3 and the imaginary part is 2.
Every real number is complex too: 7 is 7 + 0i. And 4i is 0 + 4i.
A solved problem to study
The crew's arc model is h = -x² + 4x, in Wren's tape units. Does the water ever reach height 5?
- Set the model equal to 5: -x² + 4x = 5.
- Move everything to one side and multiply by -1: x² - 4x + 5 = 0. So a = 1, b = -4, c = 5.
- Work out the discriminant: b² - 4ac = 16 - 20 = -4.
- Write the root: √(-4) = √4 × √(-1) = 2i.
- Finish the formula: x = (4 ± 2i) ÷ 2.
- Divide both parts by 2: x = 2 - 1i or x = 2 + 1i.
Both solutions have an imaginary part, so neither is a distance along the bench. In the build, that means the arc never gets there.
The equation still has two solutions. They are complex, and this week you learn to work with them.
Powers of i
| Power | Work | Value |
|---|---|---|
| i¹ | i | i |
| i² | by definition | -1 |
| i³ | i² × i = -1 × i | -i |
| i⁴ | i² × i² = -1 × -1 | 1 |
| i⁵ | i⁴ × i = 1 × i | i |
The values cycle: i, -1, -i, 1, then again. Divide the power by 4 and the remainder tells you where in the cycle you are.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| Some real number squares to -4. | ? |
| (2i)² = -4. | ? |
| 4 × 4 - 4 × 1 × 5 = -4 | ? |
| Every real number is also a complex number. | ? |
| The real part of 3 + 2i is 2. | ? |
Then write i¹⁰⁰ on the back and use the remainder rule to find its value.
A strong start. Tomorrow you add, subtract and multiply complex numbers, and compare two ways to multiply.