4. Complex Numbers and Quadratic Equations

9 topics2 h

Chapter 4 of NCERT Mathematics for Class 11: Complex Numbers and Quadratic Equations. Three sections — Why the real numbers are not enough, Doing arithmetic in the enlarged system and Complex numbers as points of a plane. Nine videos, 2 h in all.

Worked answers to the 28 exercise questions in this chapter: Exercise 4.1 · Miscellaneous Exercise

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Part 1

Why the real numbers are not enough

13 minAn equation with no real solution, and the symbol invented to solve itSquare twenty-one ordinary numbers and not one comes out negative. That is evidence, not proof. The proof is three cases — positive, negative, nothing — none below zero.13 min · 11 parts13 minSplitting a number into two parts, and when two such numbers agreeA complex number is not one quantity wearing a strange coat. It is two ordinary real numbers held in two slots, and almost every mistake made with this topic comes from forgetting that the second slot exists. Get that straight and one equation between complex expressions turns into two equations between ordinary unknowns - which is the most useful single move in the whole subject.13 min · 12 parts

Part 2

Doing arithmetic in the enlarged system

13 minAdding and subtracting componentwise, and undoing an additionEight plausible rules for adding two of these numbers, and three pass every test of order and grouping. Only one falls out of simply rearranging a + ib + c + id.13 min · 11 parts16 minMultiplying, then inverting: how division becomes possibleThe commonest way to multiply two of these numbers has two terms in it. Open the brackets properly and there are four — one of them carrying the symbol squared.16 min · 13 parts12 minWhy the powers of the new symbol run round a cycle of fourMultiply i by itself forever and nothing grows — the whole infinite list holds four numbers, repeating: i, minus one, minus i, one. Step two is not the return; step four is.12 min · 11 parts13 minSquare roots of negative numbers, and the surd rule that stops working√(−4) × √(−9) is not √36 — multiplied correctly it is 2i × 3i, which is 6i², minus six. The positive-root rule has nothing to hold once the number turns negative.13 min · 12 parts14 minWhich school algebra identities survive the enlargement, and whyThe square-of-a-sum identity was proved once, for ordinary numbers, from two laws: a product spreads across a sum, and a product can be reordered. Complex numbers keep both.14 min · 12 parts

Part 3

Complex numbers as points of a plane

16 minSize and reflection: two quantities that turn algebra into geometryMultiply three plus i by three minus i and every trace of i disappears — what remains is ten, an ordinary real number, and exactly the modulus squared.16 min · 13 parts14 minReading the plane: real axis, imaginary axis, and a mirror image for the conjugateTwo plus four i and the point two-across-four-up are not just similar. Plot forty-nine numbers by coordinates and every one lands on its own point, nothing lost.14 min · 12 parts

What you will be able to do

  • State why x² = −1 has no solution among the real numbers, giving the sign argument rather than an appeal to a calculator or a graph
  • Given a number in the form a + ib, state its real part and its imaginary part, and say that both are real numbers
  • Add two numbers written in the form a + ib and present the result in that same form
  • Derive the multiplication rule by expanding a product of two numbers in the form a + ib and substituting for i²
  • Compute i³, i⁴, i⁵ and i⁶ from the single relation i² = −1
  • List both square roots of −1 and verify each by squaring
  • State the identity the chapter proves in full, for arbitrary complex numbers
  • Compute the modulus of a given complex number and state that the value is a non-negative real number
  • Convert a complex number into a coordinate pair and back

What this chapter assumes you already know

  • The real number line, and that squaring any real number gives a result that is zero or positive
  • Solving a linear equation in one unknown, and what it means for an equation to have no solution in a stated system
  • Surd notation, and that √k for a positive real k names the non-negative root
  • That earlier school number systems have been enlarged before — whole numbers to integers, integers to rationals
  • That a real number is positive, negative or zero, and that its square is never negative
  • Solving two simultaneous linear equations in two unknowns
  • Handling improper fractions and negative coefficients confidently
  • Fluent addition and subtraction of signed fractions with unlike denominators

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "There must be some real number that squares to −1; we just haven't found it."
  • "i is imaginary, so the arithmetic built on it is make-believe."
  • "In a + ib, the b is imaginary."
  • "√(−1) is just notation, so I can do anything with it."
  • "Extending a number system means the old rules are gone."
  • "The imaginary part of 2 + i5 is 5i."
  • "Equality of complex numbers is just one equation, like for reals."
  • "You can balance a shortfall in one part against a surplus in the other."