SME
Grade 11 Math · Problem type
Powers of i
Use this for “Simplify i²⁷.”
Diagram
z = 1(cos 270° + i sin 270°) ≈ −i.
Power of i: 27
Remainder of n ÷ 4: 3
This is an example. Type your own values to start a new problem.
i
=
p
+
(
q
)
i
r
Remainder of n ÷ 4
calculated
Formulas
r
=
n
mod
4
3 = 27 mod 4
p
=
real
part
of
i
r
0
=
real
part
of
i
3
q
=
imaginary
part
of
i
r
−1
=
imaginary
part
of
i
3
Assumptions
•
The powers of i repeat every 4: i, −1, −i, 1, then i again.
•
So iⁿ = iʳ, where r is the remainder of n ÷ 4.
•
On the plane each power of i is a quarter turn more around the unit circle.
Step-by-step
WE KNOW
n = 27
FIND
remainder of n ÷ 4 (r), real part (p), imaginary part (q)
Step 1 · Find remainder of n ÷ 4 (r)
Use r = n mod 4
i
4
=
1,
so
every
4
factors
of
i
drop
out:
only
the
remainder
counts.
r
=
27
mod
4
27 = 4 × 6 + 3
r
=
3
Step 2 · Find real part (p)
Use p = real part of iʳ
Read
it
from
the
cycle
i
0
=
1,
i
1
=
i,
i
2
=
−1,
i
3
=
−i.
p
=
real
part
of
i
3
p
=
0
Step 3 · Find imaginary part (q)
Use q = imaginary part of iʳ
The number of i in the same power.
q
=
imaginary
part
of
i
3
i
27
=
(
i
4
)
6
×
i
3
=
1
×
i
3
=
−i
q
=
−1
Check
Put the numbers back in. Both sides should match.
3 = 27 mod 4 ✓
0
=
real
part
of
i
3
✓
−1
=
imaginary
part
of
i
3
✓
Topic check
See what you know: 5 questions on this topic
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