Intuition The core idea in one breathWe invented a new number i i i i 2 = − 1 i^2 = -1 i 2 = − 1 ≥ 0 \ge 0 ≥ 0 i i i i i i
Intuition Why we need a new numberThe equation x 2 = − 1 x^2 = -1 x 2 = − 1 no real solution , because squaring any real number gives something ≥ 0 \ge 0 ≥ 0 invent a symbol that solves it. Inventing new numbers to solve unsolvable equations is exactly how we got negatives (to solve x + 1 = 0 x+1=0 x + 1 = 0 2 x = 1 2x=1 2 x = 1
Definition The imaginary unitThe imaginary unit i i i i = − 1 , i 2 = − 1. i = \sqrt{-1}, \qquad i^2 = -1. i = − 1 , i 2 = − 1. complex number is anything of the form a + b i a + bi a + bi a , b a,b a , b a a a real part , b b b imaginary part .
Common mistake Steel-man: "− 1 \sqrt{-1} − 1 Why it feels right: in school x \sqrt{x} x x ≥ 0 x\ge 0 x ≥ 0 The fix: we extend the meaning. i i i purely by i 2 = − 1 i^2=-1 i 2 = − 1 i i i i 2 i^2 i 2 − 1 -1 − 1
We use only i 2 = − 1 i^2 = -1 i 2 = − 1
i 0 = 1 (any nonzero base to the 0) i^0 = 1 \quad \text{(any nonzero base to the 0)} i 0 = 1 (any nonzero base to the 0) i 1 = i i^1 = i i 1 = i i 2 = − 1 (the definition) i^2 = -1 \quad \text{(the definition)} i 2 = − 1 (the definition)
Why this step? From here we just keep multiplying by i i i i 2 i^2 i 2 − 1 -1 − 1
i 3 = i 2 ⋅ i = ( − 1 ) ⋅ i = − i i^3 = i^2 \cdot i = (-1)\cdot i = -i i 3 = i 2 ⋅ i = ( − 1 ) ⋅ i = − i
Why this step? Split off one factor of i 2 i^2 i 2 − 1 -1 − 1
i 4 = i 2 ⋅ i 2 = ( − 1 ) ( − 1 ) = 1 i^4 = i^2 \cdot i^2 = (-1)(-1) = 1 i 4 = i 2 ⋅ i 2 = ( − 1 ) ( − 1 ) = 1
Why this step? We're back to 1 1 1 multiplying by i i i
Because i 4 = 1 i^4 = 1 i 4 = 1 i 5 = i 4 ⋅ i = 1 ⋅ i = i , i 6 = i 4 ⋅ i 2 = − 1 , … i^5 = i^4 \cdot i = 1\cdot i = i,\qquad i^6 = i^4\cdot i^2 = -1,\ \dots i 5 = i 4 ⋅ i = 1 ⋅ i = i , i 6 = i 4 ⋅ i 2 = − 1 , …
Intuition Dual coding — see it as rotationMultiplying by i i i rotates a point 90 ∘ 90^\circ 9 0 ∘ in the plane. Start at 1 1 1 90 ∘ 90^\circ 9 0 ∘ i i i − 1 -1 − 1 − i -i − i 1 1 1 90 ∘ = 360 ∘ 90^\circ = 360^\circ 9 0 ∘ = 36 0 ∘ That is why the period is 4.
Worked example Example 1 — Computei 27 i^{27} i 27 Step 1: Divide the exponent by 4: 27 = 4 × 6 + 3 27 = 4\times 6 + 3 27 = 4 × 6 + 3 r = 3 r=3 r = 3 Why? Only the remainder decides the answer (from i n = i n m o d 4 i^n=i^{n\bmod 4} i n = i n mod 4 Step 2: i 27 = i 3 = − i . i^{27} = i^{3} = -i. i 27 = i 3 = − i . Answer: i 27 = − i i^{27} = -i i 27 = − i
Worked example Example 2 — Computei 100 i^{100} i 100 Step 1: 100 = 4 × 25 100 = 4\times 25 100 = 4 × 25 0 0 0 Why? 100 100 100 Step 2: i 100 = i 0 = 1. i^{100} = i^0 = 1. i 100 = i 0 = 1. Answer: 1 1 1
Worked example Example 3 — Simplifyi 15 + i 16 + i 17 + i 18 i^{15} + i^{16} + i^{17} + i^{18} i 15 + i 16 + i 17 + i 18 Step 1: Reduce each: 15 → r = 3 ⇒ − i 15\to r{=}3\Rightarrow -i 15 → r = 3 ⇒ − i 16 → r = 0 ⇒ 1 16\to r{=}0\Rightarrow 1 16 → r = 0 ⇒ 1 17 → r = 1 ⇒ i 17\to r{=}1\Rightarrow i 17 → r = 1 ⇒ i 18 → r = 2 ⇒ − 1 18\to r{=}2\Rightarrow -1 18 → r = 2 ⇒ − 1 Why? Apply the remainder rule term by term.
Step 2: Add: ( − i ) + 1 + i + ( − 1 ) = 0. (-i) + 1 + i + (-1) = 0. ( − i ) + 1 + i + ( − 1 ) = 0. Answer: 0 0 0 Insight: any four consecutive powers of i i i 0 0 0 , because they are { 1 , i , − 1 , − i } \{1,i,-1,-i\} { 1 , i , − 1 , − i } 1 + i − 1 − i = 0 1+i-1-i=0 1 + i − 1 − i = 0
Worked example Example 4 — A trap with− a − b \sqrt{-a}\,\sqrt{-b} − a − b − 4 ⋅ − 9 \sqrt{-4}\cdot\sqrt{-9} − 4 ⋅ − 9 Wrong route: ( − 4 ) ( − 9 ) = 36 = 6 \sqrt{(-4)(-9)}=\sqrt{36}=6 ( − 4 ) ( − 9 ) = 36 = 6 Right route: − 4 = 2 i , − 9 = 3 i \sqrt{-4}=2i,\ \sqrt{-9}=3i − 4 = 2 i , − 9 = 3 i = 6 i 2 = − 6 =6i^2=-6 = 6 i 2 = − 6 Why the difference? See the mistake box below.
Common mistake Steel-man: "a b = a b \sqrt{a}\sqrt{b}=\sqrt{ab} a b = ab Why it feels right: it's a true rule — for non-negative reals.
The fix: the identity a b = a b \sqrt{a}\sqrt{b}=\sqrt{ab} a b = ab fails when both a , b < 0 a,b<0 a , b < 0 . Always convert to i i i − x = i x \sqrt{-x} = i\sqrt{x} − x = i x x > 0 x>0 x > 0 before multiplying. Then − 4 − 9 = ( 2 i ) ( 3 i ) = − 6 \sqrt{-4}\sqrt{-9}=(2i)(3i)=-6 − 4 − 9 = ( 2 i ) ( 3 i ) = − 6
Common mistake Steel-man: "i 3 = i ⋅ i ⋅ i = i i^3 = i \cdot i \cdot i = i i 3 = i ⋅ i ⋅ i = i Why it feels right: people multiply i ⋅ i ⋅ i i\cdot i\cdot i i ⋅ i ⋅ i i ⋅ i = − 1 i\cdot i=-1 i ⋅ i = − 1 The fix: i ⋅ i = i 2 = − 1 i\cdot i = i^2 = -1 i ⋅ i = i 2 = − 1 i 3 = ( − 1 ) ⋅ i = − i i^3=(-1)\cdot i=-i i 3 = ( − 1 ) ⋅ i = − i not i i i
Mnemonic Remember the cycle"1, i, minus-1, minus-i" — like the four compass turns of 90 ∘ 90^\circ 9 0 ∘ E → N → W → S (East=1 1 1 i i i − 1 -1 − 1 − i -i − i "I N eed W hole S leep" for the order 1 , i , − 1 , − i 1,\,i,\,{-}1,\,{-}i 1 , i , − 1 , − i
Recall Feynman: explain to a 12-year-oldImagine a clock with only 4 numbers : 1 1 1 i i i − 1 -1 − 1 − i -i − i i i i next spot, going around. After 4 jumps you're back where you started — just like after 12 hours a clock reads the same time. So to find i i i left over after taking away full loops of 4. That leftover (0, 1, 2, or 3) tells you exactly which of the four spots you land on. And why does i 2 = − 1 i^2=-1 i 2 = − 1 decided to build a number that does that — the whole point was to answer "what times itself gives a negative?"
Recall Active recall — cover the answers
What single equation defines i i i i 2 = − 1 i^2 = -1 i 2 = − 1 i = − 1 i=\sqrt{-1} i = − 1
Why is the period of the powers 4? ::: Because i 4 = 1 i^4=1 i 4 = 1 i i i i n = i n m o d 4 i^n=i^{n\bmod 4} i n = i n mod 4
What is i 3 i^3 i 3 i i i − i -i − i i 2 = − 1 i^2=-1 i 2 = − 1 i 3 = ( − 1 ) i = − i i^3=(-1)i=-i i 3 = ( − 1 ) i = − i
What is the defining property of i i i i 2 = − 1 i^2=-1 i 2 = − 1 (so
i = − 1 i=\sqrt{-1} i = − 1 ).
Why can't i i i Every real squared is
≥ 0 \ge 0 ≥ 0 , but
i 2 = − 1 < 0 i^2=-1<0 i 2 = − 1 < 0 .
State the four values of i 4 k , i 4 k + 1 , i 4 k + 2 , i 4 k + 3 i^{4k},i^{4k+1},i^{4k+2},i^{4k+3} i 4 k , i 4 k + 1 , i 4 k + 2 , i 4 k + 3 1 , i , − 1 , − i 1,\ i,\ -1,\ -i 1 , i , − 1 , − i .
How do you compute i n i^n i n Find
r = n m o d 4 r=n\bmod 4 r = n mod 4 ; answer is
i r i^r i r from
{ 1 , i , − 1 , − i } \{1,i,-1,-i\} { 1 , i , − 1 , − i } .
Compute i 27 i^{27} i 27 27 m o d 4 = 3 ⇒ − i 27\bmod4=3\Rightarrow -i 27 mod 4 = 3 ⇒ − i .
Compute i 100 i^{100} i 100 100 m o d 4 = 0 ⇒ 1 100\bmod4=0\Rightarrow 1 100 mod 4 = 0 ⇒ 1 .
What is 1 / i 1/i 1/ i − i -i − i (multiply top & bottom by
i i i :
i / i 2 = i / ( − 1 ) i/i^2=i/(-1) i / i 2 = i / ( − 1 ) ).
Sum of any 4 consecutive powers of i i i 0 0 0 , since
{ 1 , i , − 1 , − i } \{1,i,-1,-i\} { 1 , i , − 1 , − i } sum to
0 0 0 .
Why does − 4 − 9 = − 6 \sqrt{-4}\sqrt{-9}=-6 − 4 − 9 = − 6 6 6 6 The rule
a b = a b \sqrt a\sqrt b=\sqrt{ab} a b = ab fails for two negatives; convert first:
( 2 i ) ( 3 i ) = 6 i 2 = − 6 (2i)(3i)=6i^2=-6 ( 2 i ) ( 3 i ) = 6 i 2 = − 6 .
Geometric meaning of multiplying by i i i A
90 ∘ 90^\circ 9 0 ∘ anticlockwise rotation in the complex plane.
Complex Numbers — the parent framework a + b i a+bi a + bi Argand Diagram — where the 4-cycle shows up as 90 ∘ 90^\circ 9 0 ∘ Modulus and Argument — multiplying by i i i 90 ∘ 90^\circ 9 0 ∘ Roots of Unity — { 1 , i , − 1 , − i } \{1,i,-1,-i\} { 1 , i , − 1 , − i } Euler's Formula — i = e i π / 2 i = e^{i\pi/2} i = e iπ /2 i i i π / 2 \pi/2 π /2 Solving Quadratic Equations — where negative discriminants force us to use i i i
x squared = -1 has no real solution
i = sqrt(-1), i squared = -1
Multiply by i rotates 90 degrees
Intuition Hinglish mein samjhoDekho, simple si baat hai: koi bhi real number ko square karo to answer hamesha ≥ 0 \ge 0 ≥ 0 x 2 = − 1 x^2=-1 x 2 = − 1 i i i i 2 = − 1 i^2 = -1 i 2 = − 1 i i i i 2 i^2 i 2 − 1 -1 − 1
Ab powers ka magic: i 1 = i i^1=i i 1 = i i 2 = − 1 i^2=-1 i 2 = − 1 i 3 = i 2 ⋅ i = − i i^3=i^2\cdot i=-i i 3 = i 2 ⋅ i = − i i 4 = i 2 ⋅ i 2 = ( − 1 ) ( − 1 ) = 1 i^4=i^2\cdot i^2=(-1)(-1)=1 i 4 = i 2 ⋅ i 2 = ( − 1 ) ( − 1 ) = 1 i 4 i^4 i 4 1 1 1 remainder dekho: remainder 0 → 1 1 1 i i i − 1 -1 − 1 − i -i − i i 27 i^{27} i 27 27 ÷ 4 27\div4 27 ÷ 4 3 3 3 − i -i − i
Geometry wala intuition bhi mast hai: i i i 90 ∘ 90^\circ 9 0 ∘ 1 1 1 i i i − 1 -1 − 1 − i -i − i 1 1 1 90 ∘ 90^\circ 9 0 ∘ 360 ∘ 360^\circ 36 0 ∘
Ek common galti se bachna: − 4 ⋅ − 9 \sqrt{-4}\cdot\sqrt{-9} − 4 ⋅ − 9 36 = 6 \sqrt{36}=6 36 = 6 i i i 2 i × 3 i = 6 i 2 = − 6 2i \times 3i = 6i^2 = -6 2 i × 3 i = 6 i 2 = − 6
