What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Define a complex number and name its parts.

A number \(a+ib\) with \(a,b\in\mathbb{R}\); \(a\) is the real part \(\operatorname{Re}z\) and \(b\) is the imaginary part \(\operatorname{Im}z\).

Multiply \((3+i5)(2+i6)\).

\((3\cdot2-5\cdot6)+i(3\cdot6+5\cdot2)=-24+i28\).

Find the multiplicative inverse of \(2-3i\).

\(z^{-1}=\dfrac{\bar z}{|z|^2}=\dfrac{2+3i}{13}=\dfrac{2}{13}+\dfrac{3}{13}i\), since \(|z|^2=13\).

State the cycle of powers of \(i\).

For any integer \(k\): \(i^{4k}=1,\ i^{4k+1}=i,\ i^{4k+2}=-1,\ i^{4k+3}=-i\).

When are two complex numbers equal?

\(a+ib=c+id\) iff \(a=c\) and \(b=d\), i.e. real parts equal and imaginary parts equal.

Define the modulus and conjugate of \(z=a+ib\).

The modulus is \(|z|=\sqrt{a^2+b^2}\) and the conjugate is \(\bar z=a-ib\); together \(z\bar z=|z|^2\).

Express \(\dfrac{5+2i}{1-2i}\) in the form \(a+ib\).

Multiplying by the conjugate, \(\dfrac{(5+2i)(1+2i)}{1+4}=\dfrac{3+6\sqrt2\,\text{-form}}{}=1+2\sqrt2\,i\) — i.e. it simplifies to \(1+2\sqrt2\,i\).

Why does \(\sqrt a\,\sqrt b=\sqrt{ab}\) fail for two negatives?

If \(a<0\) and \(b<0\), then \(\sqrt{-1}\,\sqrt{-1}=i^2=-1\ne\sqrt{(-1)(-1)}=1\); the identity holds only when at least one of \(a,b\) is non-negative.

What does the conjugate \(\bar z\) represent in the Argand plane?

If \(z=x+iy\) is the point \((x,y)\), then \(\bar z=x-iy\) is its mirror image \((x,-y)\) across the real axis.

State the property \(|z_1 z_2|\) and \(\overline{z_1+z_2}\).

\(|z_1 z_2|=|z_1||z_2|\) and \(\overline{z_1+z_2}=\bar z_1+\bar z_2\).

Complex Numbers and Quadratic Equations

Q: Why does \(\sqrt a\,\sqrt b=\sqrt{ab}\) fail for two negatives?

If \(a<0\) and \(b<0\), then \(\sqrt{-1}\,\sqrt{-1}=i^2=-1\ne\sqrt{(-1)(-1)}=1\); the identity holds only when at least one of \(a,b\) is non-negative.

Q: What does the conjugate \(\bar z\) represent in the Argand plane?

If \(z=x+iy\) is the point \((x,y)\), then \(\bar z=x-iy\) is its mirror image \((x,-y)\) across the real axis.

Q: State the property \(|z_1 z_2|\) and \(\overline{z_1+z_2}\).

\(|z_1 z_2|=|z_1||z_2|\) and \(\overline{z_1+z_2}=\bar z_1+\bar z_2\).

Class 11 Mathematics · Chapter 4

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Every formula of Ch 1–14, with its condition

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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

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Summary

The real number system is extended so that equations like \(x^2+1=0\) have solutions. Defining \(i=\sqrt{-1}\) with \(i^2=-1\), a complex number has the form \(z=a+ib\) where \(a=\operatorname{Re}z\) and \(b=\operatorname{Im}z\). Two complex numbers are equal iff their real and imaginary parts match. Addition, subtraction and multiplication follow term-by-term rules, e.g. \((a+ib)(c+id)=(ac-bd)+i(ad+bc)\), and every non-zero \(z\) has a multiplicative inverse \(z^{-1}=\bar z/|z|^2\). Powers of \(i\) cycle: \(i^{4k}=1,\ i^{4k+1}=i,\ i^{4k+2}=-1,\ i^{4k+3}=-i\). The modulus \(|z|=\sqrt{a^2+b^2}\) and conjugate \(\bar z=a-ib\) satisfy \(z\bar z=|z|^2\). Addition obeys the closure, commutative, associative laws, with additive identity \(0+i0\) and additive inverse \(-z\); multiplication has identity \(1+i0\) and is distributive over addition. The square root of a negative number \(-a\) (with \(a>0\)) is \(\sqrt a\,i\), but the rule \(\sqrt a\,\sqrt b=\sqrt{ab}\) fails when both are negative. Geometrically a complex number \(x+iy\) corresponds to the point \((x,y)\) in the Argand (complex) plane, where the real axis and imaginary axis replace the coordinate axes, \(|z|\) is the distance from the origin and \(\bar z\) is the mirror image of \(z\) in the real axis. Useful properties include \(|z_1 z_2|=|z_1||z_2|\) and \(\overline{z_1\pm z_2}=\bar z_1\pm\bar z_2\). These tools provide solutions to quadratic equations whose discriminant is negative.

The imaginary unit and complex numbersEquality, real and imaginary partsAlgebra of complex numbersPowers of \(i\) and square roots of negativesModulus and conjugateArgand plane and geometric representation

Key terms

Imaginary unit

\(i=\sqrt{-1}\), so that \(i^2=-1\).

Complex number

\(z=a+ib\) with real part \(a=\operatorname{Re}z\) and imaginary part \(b=\operatorname{Im}z\).

Conjugate

\(\bar z=a-ib\), the mirror image of \(z\) in the real axis.

Modulus

\(|z|=\sqrt{a^2+b^2}\), the distance of \(z\) from the origin; \(z\bar z=|z|^2\).

Multiplicative inverse

\(z^{-1}=\dfrac{a-ib}{a^2+b^2}=\dfrac{\bar z}{|z|^2}\) for \(z\ne 0\).

Argand plane

The plane in which \(z=x+iy\) is plotted as the point \((x,y)\).

Extra questions & answers

Try one from this chapter

The value of \(i^2\) is:

Flashcards & interactive practice