Complex Numbers

Separation into real and imaginary parts

Separating a complex expression into its real and imaginary parts involves algebraic manipulation using conjugates, De Moivre's theorem, or logarithmic/trigonometric identities to express it uniquely in the form $u + iv$.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Fundamental Principle of Complex Decomposition

Let $z = x + iy$ be a complex number where $x, y \in \mathbb{R}$. Any well-defined complex-valued function $f(z)$ can be uniquely decomposed into a real part $u(x, y)$ and an imaginary part $v(x, y)$ such that $f(x + iy) = u(x, y) + iv(x, y)$, where $u, v: \mathbb{R}^2 \to \mathbb{R}$ are real-valued scalar functions.

📐 Geometric Interpretation

Geometrically, separation maps points from the complex z-plane into a two-variable real coordinate system representing orthogonal Cartesian components of the function value.

⚙️ Algebraic & Structural Meaning

It establishes a bridge between complex algebra and multivariable real calculus, enabling the application of techniques like partial differentiation, harmonic conjugates, and Cartesian coordinate transformations.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Algebraic Domain Restriction	`Entire Complex Plane \mathbb{C} \setminus \{ \text{singularities} \}`	Strictly Mandatory / Necessary	Division by zero or encountering branch cuts (such as in logarithmic or fractional power functions) leads to undefined or multivalued real/imaginary parts.
Denominator Non-Zero Condition	`c + id \neq 0`	Strictly Mandatory	Rational complex expressions cannot be rationalized using the complex conjugate if the denominator vanishes.
Principal Branch Selection	`(-\pi, \pi] for arguments`	Essential for conclusion	Multivalued functions like $\log(z)$ or $z^n$ will yield ambiguous real/imaginary values if the principal branch is not explicitly specified.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Rationalization of Complex Denominators

$$\frac{a + ib}{c + id} = \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2}$$

Rule: Immediately multiply numerator and denominator by the complex conjugate of the denominator $(c - id)$ to instantly isolate real and imaginary coefficients.

Application: Directly tested in UP PGT/TGT to simplify powers of complex fractions.

Euler's Form for Powers and Roots

$$(x + iy)^n = r^n (\cos n\theta + i \sin n\theta), \quad \text{where } r = \sqrt{x^2+y^2}, \theta = \tan^{-1}\left(\frac{y}{x}\right)$$

Rule: Convert algebraic complex numbers into polar/exponential form before applying exponents or extracting roots.

Application: Frequent in DSSSB PGT for evaluating expressions like $(1 + i)^n + (1 - i)^n$.

Separation of Logarithmic Complex Numbers

$$\log(x + iy) = \frac{1}{2}\ln(x^2 + y^2) + i \tan^{-1}\left(\frac{y}{x}\right)$$

Rule: Use the standard logarithmic expansion where the real part is half the natural log of the squared modulus, and the imaginary part is the principal argument.

Trigonometric Functions of Complex Arguments

$$\sin(x + iy) = \sin x \cosh y + i \cos x \sinh y$$

Rule: Apply hyperbolic identities: $\sin(ix) = i \sinh x$ and $\cos(ix) = \cosh x$ to separate sine and cosine of complex sums instantly.

Application: Tested in LT Grade and UP PGT to find real parts of $\sin(a + ib)$.

Inverse Trigonometric Real/Imaginary Extraction

$$ \text{If } \sin^{-1}(x + iy) = \alpha + i\beta, \text{ then equate real and imaginary parts after taking sine on both sides.} $$

Rule: Convert inverse trigonometric equations into standard trigonometric forms using expansion formulas for $\sin(A+B)$.

Application: Frequently asked in UP PGT for separation of $\tan^{-1}(x + iy)$.

Rapid Elimination via Modulus Check

$$ |u + iv| = |f(z)| = \frac{|Numerator|}{|Denominator|} $$

Rule: Check the magnitude of options. If an option does not satisfy the computed modulus squared, eliminate it instantly.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Ignoring Principal Branch Multi-Valuedness

Common Pitfall: Treating logarithmic and fractional power separations as single-valued without considering the $2k\pi$ periodicity of arguments.

Counterexample: \log(1) = 2k\pi i \quad (k \in \mathbb{Z})

Why it fails: Forgetting the general argument adds incorrect imaginary components, leading to wrong option selection.

🛡️ Exam Defense: Always check whether the question specifies the 'principal value' or general value.

⚠️ COMMON ERROR #02

Hyperbolic vs Trigonometric Sign Confusion

Common Pitfall: Mixing up signs when expanding $\cos(x + iy) = \cos x \cosh y - i \sin x \sinh y$, particularly missing the negative sign.

Counterexample: \cos(x + iy) \neq \cos x \cosh y + i \sin x \sinh y

Why it fails: Leads to choosing the sign-inverted distractor deliberately placed by the examiner.

🛡️ Exam Defense: Memorize the fundamental expansion: $\cos(A+B) = \cos A \cos B - \sin A \sin B$ combined with $\cos(iy) = \cosh y$ and $\sin(iy) = i \sinh y$.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Algebraic Rationalization	Denominator $\neq 0$	$\frac{x+iy}{a+ib} = \frac{(ax+by) + i(ay-bx)}{a^2+b^2}$	Orthogonal projection in Cartesian plane	Fractions with complex numbers in denominator
Polar / De Moivre Form	$r > 0$	$r^n(\cos n\theta + i \sin n\theta)$	Scaling by $r^n$ and rotation by $n\theta$	High powers or roots like $(1+i)^n$
Complex Logarithm	$z \neq 0$	$\frac{1}{2}\ln(x^2+y^2) + i\arg(z)$	Mapping magnitude to radius and angle to arc	Logarithmic expressions with complex variables
Hyperbolic Separation	Standard domain	$\sin(x+iy) = \sin x \cosh y + i \cos x \sinh y$	Combination of circular and hyperbolic orbits	Trigonometric functions of complex sums

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $\sin^{-1}(x + iy) = \alpha + i\beta$, where $x, y, \alpha, \beta$ are real, find the value of $x^2 + y^2$ or separate into real and imaginary parts.

(A) $\sin^2 \alpha + \sinh^2 \beta$(B) $\cos^2 \alpha + \cosh^2 \beta$(C) $\sin^2 \alpha - \sinh^2 \beta$(D) $\cosh^2 \alpha + \sinh^2 \beta$

📚 Method A: Formal Textbook Derivation

Step 1: Write $x + iy = \sin(\alpha + i\beta) = \sin\alpha \cos(i\beta) + \cos\alpha \sin(i\beta) = \sin\alpha \cosh\beta + i\cos\alpha \sinh\beta$. Step 2: Equate real and imaginary parts: $x = \sin\alpha \cosh\beta$ and $y = \cos\alpha \sinh\beta$. Step 3: Use $\cosh^2\beta - \sinh^2\beta = 1$ to eliminate $\beta$ or compute $x^2 + \dots$

⚡ Method B: 25-Second Exam Shortcut

Recognize the standard result for inverse hyperbolic-trigonometric links: real part squared terms pair with hyperbolic sine/cosine identities.

💡 Tip: Direct standard result frequently asked verbatim in UP PGT.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Separate $\tan(x + iy)$ into real and imaginary parts.

(A) $\frac{\sin 2x + i \sinh 2y}{\cos 2x + \cosh 2y}$(B) $\frac{\sin 2x - i \sinh 2y}{\cos 2x - \cosh 2y}$(C) $\frac{\sinh 2x + i \sin 2y}{\cosh 2x + \cos 2y}$(D) None of these

📚 Method A: Formal Textbook Derivation

Step 1: Write $\tan(x + iy) = \frac{\sin(x + iy)}{\cos(x + iy)}$. Step 2: Multiply numerator and denominator by the complex conjugate of the denominator $\cos(x - iy)$. Step 3: Simplify using product-to-sum trigonometric and hyperbolic identities to obtain the result.

⚡ Method B: 25-Second Exam Shortcut

Remember numerator has $\sin 2x + i \sinh 2y$ and denominator has $\cos 2x + \cosh 2y$.

💡 Tip: Watch out for sign flips between $\sin$ and $\sinh$ terms in options.

Practice questions

Choose an answer to check it and see the solution:

The real part of $\frac{1}{1 + i}$ is:

A $1$

B $1/2$

C $-1/2$

D $0$

Derivation & Analysis:

Multiply numerator and denominator by $1 - i$: $\frac{1 - i}{(1+i)(1-i)} = \frac{1-i}{2} = \frac{1}{2} - \frac{i}{2}$. The real part is $1/2$.

If $x + iy = \frac{1}{1 + \cos \theta + i \sin \theta}$, then the value of $x$ is:

A $1/2$

B $1$

C $\cos \theta$

D $2 \cos(\theta/2)$

Derivation & Analysis:

Rationalize the denominator using $1 + \cos\theta - i\sin\theta$. Denominator becomes $(1+\cos\theta)^2 + \sin^2\theta = 2 + 2\cos\theta = 4\cos^2(\theta/2)$. Real part works out to $1/2$.

The imaginary part of $\log(3 + 4i)$ is:

A $\tan^{-1}(3/4)$

B $\tan^{-1}(4/3)$

C $\ln(5)$

D $4/3$

Derivation & Analysis:

By definition, $\log(x + iy) = \frac{1}{2}\ln(x^2+y^2) + i\tan^{-1}(y/x)$. Here $x=3, y=4$, so imaginary part is $\tan^{-1}(4/3)$.

If $\cos(x + iy) = A + iB$, then $A$ is equal to:

A $\cos x \cosh y$

B $-\sin x \sinh y$

C $\cos x \cos y$

D $\sin x \sinh y$

Derivation & Analysis:

$\cos(x + iy) = \cos x \cos(iy) - \sin x \sin(iy) = \cos x \cosh y - i \sin x \sinh y$. Thus, real part $A = \cos x \cosh y$.

The real part of $(1 + i)^n$ when $n = 4$ is:

A $-4$

B $4$

C $0$

D $2$

Derivation & Analysis:

$(1+i)^2 = 2i$, so $(1+i)^4 = (2i)^2 = -4$. The real part is $-4$.

If $u + iv = e^{x + iy}$, then $v$ is equal to:

A $e^x \cos y$

B $e^x \sin y$

C $e^y \cos x$

D $e^y \sin x$

Derivation & Analysis:

$e^{x+iy} = e^x \cdot e^{iy} = e^x(\cos y + i \sin y) = e^x \cos y + i e^x \sin y$. Hence $v = e^x \sin y$.

The value of $\left(\frac{1 + i}{1 - i}\right)^3$ is equal to:

A $i$

B $-i$

C $1$

D $-1$

Derivation & Analysis:

$\frac{1+i}{1-i} = i$. Therefore, $\left(\frac{1+i}{1-i}\right)^3 = i^3 = -i$.

If $\tan(x + iy) = u + iv$, then $u$ is:

A $\frac{\sin 2x}{\cos 2x + \cosh 2y}$

B $\frac{\sinh 2y}{\cos 2x + \cosh 2y}$

C $\frac{\sin 2x}{\cos 2x - \cosh 2y}$

D $\frac{\sinh 2x}{\cos 2x + \cosh 2y}$

Derivation & Analysis:

From standard separation formulas for complex tangent, the real part $u = \frac{\sin 2x}{\cos 2x + \cosh 2y}$.

The modulus of the complex number $z = 3 - 4i$ is:

A $7$

B $5$

C $25$

D $-5$

Derivation & Analysis:

$|3 - 4i| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Q10

If $z = x + iy$ and $z^2 = i$, then the values of $(x, y)$ are:

A $(\pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}})$

B $(\pm 1, \pm 1)$

C $(\pm \frac{1}{\sqrt{2}}, \mp \frac{1}{\sqrt{2}})$

D $(1, 1)$

Derivation & Analysis:

$z^2 = x^2 - y^2 + 2ixy = i$. Equating parts: $x^2 - y^2 = 0 \implies x = \pm y$. Also $2xy = 1$. Solving gives $x = \pm \frac{1}{\sqrt{2}}, y = \pm \frac{1}{\sqrt{2}}$.

Common questions

Because Euler's relations connect imaginary exponents to trigonometric functions via $e^{iy} = \cos y + i \sin y$ and $e^{-iy} = \cos y - i \sin y$, leading directly to hyperbolic definitions when the argument is purely imaginary.

Forgetting that $\log(z)$ is a multi-valued function in complex analysis. Unless the principal value is explicitly requested, one must include the $2k\pi i$ periodic term in the imaginary part.

Yes. Memorizing the numerator and denominator structures for $\tan(x+iy)$ and $\tanh(x+iy)$ saves crucial minutes during high-pressure competitive exams.

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