You already know the one rule (see the parent / Complex Numbers ): i 2 = − 1 1 , i , − 1 , − i 4 drill hall . We line up every kind of question this topic can throw at you, then knock each one down with a full derivation.
Intuition Read this firstAlmost every "compute a power of i find the remainder when the exponent is divided by 4 . Everything below is just that move dressed up in different clothes — big exponents, negative exponents, sums, square roots of negatives, and word problems. Learn to spot the clothing.
Here is the full space of cases. Each later example is tagged with the cell it covers.
Cell
Case class
What makes it tricky
Example
A Small positive exponent
Nothing — warm-up
Ex 1
B Large positive exponent
Must reduce mod 4
Ex 2
C Exponent ≡ 0 ( mod 4 )
Answer is 1
Ex 2
D Negative exponent
Cycle runs backwards
Ex 3
E Sum of consecutive powers
Cancellation to 0
Ex 4
F − a − b The naive ab fails
Ex 5
G Mixed sign: − a + b
Rule holds here — contrast with F
Ex 5b
H Geometric / rotation (real-world compass)
See it as 9 0 ∘
Ex 6
I Exam twist: solve i n =
Reverse the cycle
Ex 7
J Very large exponent + algebra
Combine mod-4 with simplification
Ex 8
Worked example Example 1 (Cell A) — Computei 3 Forecast: guess the answer before reading. Is it i − i 1 − 1
Step 1. Split off one i 2 i 3 = i 2 ⋅ i Why this step? We only ever know i 2 = − 1 i 2 Step 2. Substitute i 2 = − 1 i 3 = ( − 1 ) ⋅ i = − i Why this step? Replacing i 2 − 1 only algebra i
Answer: i 3 = − i Verify: the mnemonic order is 1 , i , − 1 , − i 3 i 0 0 − i
Worked example Example 2 (Cells B, C) — Computei 2023 i 2024 Forecast: two neighbouring exponents. Do they give neighbouring cycle values?
Step 1. Divide 2023 4 2023 = 4 × 505 + 3 r = 3 Why this step? Only the remainder decides the value.
Step 2. i 2023 = i 3 = − i
Step 3. Now 2024 = 4 × 506 + 0 r = 0 degenerate cell C (exponent a multiple of 4).
Why this step? A multiple of 4 means we've done whole loops only, landing back at the start.
Step 4. i 2024 = i 0 = 1
Answer: i 2023 = − i i 2024 = 1 Verify: consecutive exponents differ by one factor of i i 2024 = i ⋅ i 2023 = i ⋅ ( − i ) = − i 2 = 1
Worked example Example 3 (Cell D) — Computei − 7 Forecast: negative exponents mean reciprocals. Which of the four values?
Step 1. Recall the backward step i − 1 = − i i 1 ⋅ i i = i 2 i = − 1 i = − i Why this step? Going back one step in the cycle is multiplying by i − 1 = − i Step 2. Reduce the exponent mod 4, keeping the sign. − 7 = 4 × ( − 2 ) + 1 4 × ( − 2 ) = − 8 − 8 + 1 = − 7 r = 1 Why this step? The formula i n = i n mod 4 n r { 0 , 1 , 2 , 3 } Step 3. i − 7 = i 1 = i
Answer: i − 7 = i Verify (independent route): i − 7 = i 7 1 i 7 = i 4 ⋅ i 3 = 1 ⋅ ( − i ) = − i i − 7 = − i 1 = − i 1 ⋅ i i = − i 2 i = 1 i = i
Common mistake Steel-man: "− 7 mod 4 = 3 − i Why it feels right: you might compute 7 mod 4 = 3 The fix: remainders are always taken so 0 ≤ r < 4 − 7 not exceeding − 7 − 8 r = − 7 − ( − 8 ) = 1
Worked example Example 4 (Cell E) — Computei 50 + i 51 + i 52 + i 53 Forecast: four consecutive powers. Guess the sum before computing.
Step 1. Reduce each exponent mod 4: 50 → 2 51 → 3 52 → 0 53 → 1 Why this step? Term-by-term remainder rule.
Step 2. Substitute: i 50 = − 1 , i 51 = − i , i 52 = 1 , i 53 = i Step 3. Add: ( − 1 ) + ( − i ) + 1 + i = 0 Why this step? The four terms are exactly { 1 , i , − 1 , − i } 0
Answer: 0 Verify: factor out the smallest power: i 50 ( 1 + i + i 2 + i 3 ) = i 50 ( 1 + i − 1 − i ) = i 50 ⋅ 0 = 0 0
Worked example Example 5 (Cell F) — Compute− 16 ⋅ − 25 Forecast: is it + 20 − 20
Step 1. Convert each radical to i first : − 16 = 4 i − 25 = 5 i Why this step? The rule a b = ab i before multiplying sidesteps the trap.
Step 2. Multiply: ( 4 i ) ( 5 i ) = 20 i 2 = 20 ( − 1 ) = − 20
Answer: − 20 Wrong route to avoid: ( − 16 ) ( − 25 ) = 400 = 20 Verify: ( 4 i ) ( 5 i ) = 20 i 2 = − 20 ( − 20 ) i 2 − 1
Worked example Example 5b (Cell G) — Compute− 16 ⋅ 25 Forecast: does the sign flip like in 5, or not?
Step 1. − 16 = 4 i 25 = 5 Why this step? Only the negative radical needs the i Step 2. Multiply: ( 4 i ) ( 5 ) = 20 i
Answer: 20 i Verify: with only one negative under a root, the identity − a b = i a b = i ab does hold. Here i 16 ⋅ 25 = i 400 = 20 i both were negative and the rule failed.
Worked example Example 6 (Cell H) — A compass droneA drone starts facing East , represented by the number 1 i 9 0 ∘ 11 commands, which way does it face, and what number represents it?
Forecast: East, North, West, or South?
Step 1. Facing after 11 turns = i 11 1 Why this step? Each command multiplies by i i 11 Step 2. Reduce: 11 = 4 × 2 + 3 r = 3 i 11 = i 3 = − i Why this step? Two full 36 0 ∘ 3 Step 3. Read the figure: − i South .
Answer: the drone faces South , represented by − i Verify: 11 = 11 × 9 0 ∘ = 99 0 ∘ 990 − 720 = 27 0 ∘ Argand Diagram and Modulus and Argument for the rotation viewpoint.
Worked example Example 7 (Cell I) — Find all integersn 0 ≤ n ≤ 20 i n = − 1 Forecast: how many solutions in that range — a handful, or dozens?
Step 1. i n = − 1 r = 2 n ≡ 2 ( mod 4 ) Why this step? − 1 2 2 Step 2. List n [ 0 , 20 ] n = 2 , 6 , 10 , 14 , 18 Why this step? Start at 2 4 20
Answer: n ∈ { 2 , 6 , 10 , 14 , 18 } Verify: each of these minus 2 is divisible by 4: 0 , 4 , 8 , 12 , 16 i 2 = − 1
Worked example Example 8 (Cell J) — Simplifyi 98 i 99 + i 100 Forecast: will it be a clean single value?
Step 1. Reduce each power: 99 → 3 ⇒ − i 100 → 0 ⇒ 1 98 → 2 ⇒ − 1 Why this step? Handle every power via the remainder rule before combining.
Step 2. Substitute: − 1 ( − i ) + 1 = − 1 1 − i = − ( 1 − i ) = − 1 + i Why this step? Now it's ordinary arithmetic of complex numbers a + bi
Answer: − 1 + i Verify (algebra route): factor i 98 i 98 ( i + i 2 ) = i + i 2 = i − 1 = − 1 + i i 98
Recall Feynman: what did we actually learn across all cells?Every single problem here — big powers, negatives, roots, drones, exam reversals — collapsed to one trick: strip away full loops of four, keep the leftover. For − a i before you multiply." For geometry, the same leftover is a count of 9 0 ∘
Recall Active recall — cover the answers
i 2023 2023 mod 4 = 3 ⇒ − i i − 7 1 ⇒ i 1/ i 7 − 16 − 25 − 20 4 i , 5 i Which n [ 0 , 20 ] i n = − 1 2 , 6 , 10 , 14 , 18
i 98 i 99 + i 100 − 1 + i
