Complex number a+bi — real part, imaginary part

3.5.2 · Maths › Complex Numbers

1. Definition

KAUN sa number kya hai?

$z$	$Re (z)$	$Im (z)$	Name
$3 + 4 i$	$3$	$4$	complex
$- 7$	$- 7$	$0$	purely real
$5 i$	$0$	$5$	purely imaginary
$0$	$0$	$0$	zero

2. $i$ aata kahaan se hai? (Scratch se derivation)

HOW hum system banate hain, step by step:

Shuru karo reals $R$ se. Problem: $x^{2} = - 1$ ka koi real solution nahi kyunki $x^{2} \geq 0$ hamesha hota hai. Yeh step kyun? Hum exactly identify karte hain ki kya missing hai.
Postulate karo ek naya symbol $i$ jisme $i^{2} = - 1$ ho. Kyun? Yeh minimal naya rule hai jo gap fix karta hai.
System ko close karo $+, -, \times$ ke under. Reals aur $i$ ka koi bhi sum/product expressible rehna chahiye. Lo $a + bi$ . Yeh step kyun? Hume arithmetic chahiye jo hume set se bahar na phenke.
Multiplication ka closure check karo: $(a + bi) (c + d i) = a c + a d i + b c i + b d i^{2} = (a c - b d) + (a d + b c) i .$ Yeh step kyun? Result phir se $A + B i$ form mein hai — set closed hai, toh $C$ consistent hai.

3. Equality aur geometric picture

Har $z = a + bi$ plane mein ek point $(a, b)$ hai — Argand diagram. Horizontal axis real hai, vertical axis imaginary hai.

Figure — Complex number a+bi — real part, imaginary part

5. Feynman & memory aids

Recall Ek 12-saal ke bachche ko explain karo (click to reveal)

Normal numbers ek seedhi line par rehte hain (ek ruler). Lekin kuch puzzles, jaise "kaunsa number apne aap se multiply hokar $- 1$ deta hai?", ka koi answer us ruler par nahi hota. Toh hum ek bilkul naya direction invent karte hain — upar, ruler se hat ke — aur ek kadam upar ko " $i$ " kehte hain. Ab numbers ek poori flat sheet par rehte hain: daayein/baayein jao ( $a$ ) aur upar/neeche jao ( $b$ ). Ek complex number $a + bi$ bas kehta hai "daayein $a$ kadam chalo aur upar $b$ kadam chalo." Rule $i^{2} = - 1$ ka matlab hai: $i$ se do baar turn karna = आधा chakkar = peeche ki taraf face karna (yani $- 1$ direction).

6. Connections

Argand diagram & modulus — $a + bi$ ko $(a, b)$ ke roop mein plot karna, length $∣ z ∣ = a^{2} + b^{2}$ .
Complex conjugate — $\overset{z}{ˉ} = a - bi$ imaginary part flip karta hai.
Addition and multiplication of complex numbers — upar derive kiye gaye arithmetic rules.
Polar form and Euler's formula — $z = r (cos θ + i sin θ)$ .
Quadratic equations with complex roots — jahaan $i$ pehli baar kaam aata hai.

Imaginary unit

i

ki defining property kya hai?

i^{2} = - 1

z = a + bi

mein

Re (z)

kya hai?

a

(real coefficient)

z = a + bi

mein

Im (z)

kya hai?

b

, ek real number (na ki

bi

)

a + bi = c + d i

se kya imply hota hai?

a = c

aur

b = d

(real & imaginary parts alag-alag match karte hain)

i^{3}

simplify karo.

- i

i^{2023}

simplify karo.

- i

(kyunki

2023 mod 4 = 3

)

(a + bi) (c + d i)

A + B i

form mein likho?

(a c - b d) + (a d + b c) i

i

ek real number kyun nahi ho sakta?

Ek real number ka square

\geq 0

hota hai, lekin

i^{2} = - 1 < 0

hai.

- 4 \cdot - 9

sahi tarike se compute karo.

2 i \cdot 3 i = 6 i^{2} = - 6

Saare complex numbers ke set ka naam kya hai?

C

Complex number a+bi — real part, imaginary part

1. Definition

2. $i$ aata kahaan se hai? (Scratch se derivation)

3. Equality aur geometric picture

4. Worked examples

5. Feynman & memory aids

6. Connections

Concept Map

1. Definition

2. i aata kahaan se hai? (Scratch se derivation)

3. Equality aur geometric picture

4. Worked examples

5. Feynman & memory aids

6. Connections

Concept Map

2. $i$ aata kahaan se hai? (Scratch se derivation)