NDA Maths · Complex Numbers

Powers of i, De Moivre & Roots

The four-step cycle of powers of i, De Moivre's theorem for raising complex numbers to powers, and finding square/nth roots of a complex number.

All Complex Numbers notes NDA Maths strategy

Why this matters

Powers-of-i questions are fast marks once you use the period-4 cycle, and De Moivre turns an ugly (1+i)ⁿ computation into one angle multiplication.

Concept 1 of 2

Powers of i (the period-4 cycle)

Intuition

Powers of

i

repeat every four:

i,-1,-i,1,

then back to

i

. So any

i^n

is decided by

n \bmod 4

, and any block of four consecutive powers sums to zero — which collapses long sums instantly.

Definition

$i^1=i,\;i^2=-1,\;i^3=-i,\;i^4=1$ ; thereafter $i^n=i^{\,n\bmod 4}$ . Block sum: $i^k+i^{k+1}+i^{k+2}+i^{k+3}=0$ for any $k$ . So $\sum$ of $i^n$ over a full set of consecutive 4 is 0 — only the leftover terms survive.

Worked example

Evaluate

i^{1000}+i^{1001}+i^{1002}+i^{1003}

Four consecutive powers of $i$ sum to 0.

Answer:

0

Practice this conceptself-check · 4 quick reps

Try it yourself

What is

i^{2026}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$i^3=$ ?
2.
$i^n$ depends on?
3.
$i^k+i^{k+1}+i^{k+2}+i^{k+3}=$ ?
4.
$i^{102}$ ?

From the bank · past-year question

Example 1Complex NumbersEASY

The value of

i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3}

, where

i = \sqrt{-1}

, is

[Q3 · Apr · 2017]

Drill 9 more on powers of i (the period-4 cycle)

Concept 2 of 2

De Moivre's theorem and roots

Intuition

To raise a complex number to a power, put it in polar form and multiply the angle:

(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta

. Running it backwards gives the

n

nth-roots, equally spaced around a circle. Square roots of

a+ib

can also be found by solving

(x+iy)^2=a+ib

Definition

De Moivre: $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ (also for the modulus: $z^n=r^n e^{in\theta}$ ). nth roots of $re^{i\theta}$ : $r^{1/n}e^{i(\theta+2k\pi)/n}$ , $k=0,\ldots,n-1$ — $n$ points on a circle of radius $r^{1/n}$ . Square root of $a+ib$ : set $(x+iy)^2=a+ib$ , match parts ( $x^2-y^2=a$ , $2xy=b$ ).

Worked example

Use De Moivre to find

(1+i)^4

$1+i=\sqrt2\,e^{i\pi/4}$ , so $(1+i)^4=(\sqrt2)^4 e^{i\pi}=4(\cos\pi+i\sin\pi)$ .
$=4(-1)=-4$ .

Answer:

-4

Practice this conceptself-check · 4 quick reps

Try it yourself

Find a square root of

-i

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
De Moivre: $(\cos\theta+i\sin\theta)^n=$ ?
2.
How many distinct nth roots does a complex number have?
3.
nth roots are spaced how, geometrically?
4.
$(1+i)^4=$ ?

From the bank · past-year question

Example 2Complex NumbersHARD

Which one of the following is a square root of

\sqrt{-\sqrt{-1}}

[Q12 · Sep · 2023]

Drill 4 more on de moivre's theorem and roots

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Complex NumbersEASY

i=\sqrt{-1}

, then how many values does

i^{-2n}

have for different

n\in\mathbb{Z}

[Q48 · Sep · 2021]

Example 2Complex NumbersMODERATE

Which one of the following is a square root of

2a+2\sqrt{a^{2}+b^{2}}

, where

a,b\in\mathbb{R}

[Q15 · Sep · 2021]

Example 3Complex NumbersHARD

What is

\sum_{n=2}^{11}(i^n + i^{n+1})

, where

i = \sqrt{-1}

[Q35 · Apr · 2018]

Example 4Complex NumbersHARD

Let

\alpha

and

\beta

be real numbers and

z

be a complex number. If

z^2 + \alpha z + \beta = 0

has two distinct non-real roots with

\text{Re}(z) = 1

, then it is necessary that

[Q5 · Apr · 2018]

Example 5Complex NumbersEASY

The smallest positive integer

n

for which

\left(\dfrac{1+i}{1-i}\right)^n = 1

, is

[Q21 · Sep · 2017]

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15 questions from the bank — paginated, with cart and Word-export support.

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