Popular Courses
Beginner – Linear Regression
Introduction
Beginner – Decision Trees
Introduction
Beginner – Neural Networks
Introduction

Sign in to Curvingo

or

New Account     Reset Password     

Plans

Details
Complex Numbers and Quadratic Equations
Introduction
Complex Numbers
Algebra of Complex Numbers
The Modulus and the Conjugate of a Complex Number
Practice: Complex Number Arithmetic
Argand Plane and Polar Representation
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
Curvingo · Made in India

Share

Send us Feedback

Please let us know if you have any feedback and suggestions, or if you find any errors and bugs in our content.

Sorry, your message couldn’t be submitted. Please try again!

Thanks for your feedback!

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Complex Numbers and Quadratic EquationsIntroduction

Reading time: ~15 min

Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics. – Gauss

Riya's Dadi wants a small kitchen garden in the courtyard: tulsi, mint and a few chilli plants. She has exactly 20 m of jute rope to fence it, and she wants the rectangle inside the rope to cover 40 m². Riya grabs a notebook. One side 4 m means the other is 6 m, and 4 × 6 = 24. Too small. Surely a better pair of sides is out there?

You are Riya. The rope always goes all the way round, so the two sides always add up to 10 m. Drag one side and watch the area. Can you reach 40 m²? If not, find the biggest patch this rope can fence.

One side of the garden: ${s} m

The best this rope can do is a 5 m × 5 m square of 25 m², far short of 40. Algebra agrees. Call one side x; the other is 10 - x, so x(10 - x) = 40, which rearranges to x^2 - 10x + 40 = 0. Its discriminant is D = (-10)^2 - 4 \times 1 \times 40 = , so the equation has .

In Class 10 that was the end of the story: "no real roots" meant "no answer". This chapter is about refusing to stop there.

Cardano's Daring Move

In 1545 the Italian mathematician Gerolamo Cardano met the very same puzzle, dressed differently: find two numbers whose sum is 10 and whose product is 40. Instead of giving up, he pushed the quadratic formula through anyway:

x = \dfrac{10 \pm \sqrt{-60}}{2} = 5 \pm \sqrt{-15}

Nobody knew what \sqrt{-15} meant. But let us test his two "numbers" with one single rule: \left(\sqrt{-15}\right)^2 = -15.

Their sum is \left(5 + \sqrt{-15}\right) + \left(5 - \sqrt{-15}\right) = .

Their product has the shape (p + q)(p - q) = p^2 - q^2, so it is 25 - \left(\sqrt{-15}\right)^2 = 25 - (-15) = .

Both conditions hold! Cardano dismissed these numbers as useless and moved on, yet the arithmetic worked perfectly. All it needed was a number whose square is negative.

The simplest equation that asks for such a number is x^2 + 1 = 0, that is, x^2 = -1. The square of every real number is , so no real number solves it. We must extend the real numbers to a larger system in which x^2 = -1 has a solution. The real aim is bigger still: to solve every quadratic equation ax^2 + bx + c = 0, including those with D = b^2 - 4ac < 0, which have no solution among the real numbers.

Why invent a new kind of number?

Mathematics has grown like this before. 7 - 10 had no answer until negative numbers arrived, and 3 \div 4 had none until fractions arrived. Each time we kept everything that already worked and added the missing piece. Complex numbers are the next step: the real numbers, plus one new number whose square is −1.

From "Useless" to Essential

The trouble with square roots of negative numbers was noticed long before Cardano. Around the year 850 the Indian mathematician Mahavira wrote in his Ganita Sara Sangraha that a negative quantity is not a square, so it has no square root. Bhaskara II repeated the warning in his Bijaganita of 1150.

After Cardano, Albert Girard (around 1625) accepted square roots of negative numbers, and saw that with them a polynomial equation can have as many roots as its degree. Euler introduced the symbol i for \sqrt{-1}. Around 1830 W. R. Hamilton described the complex number a + ib as an ordered pair of real numbers (a, b): a purely mathematical definition that took the mystery out of "imaginary" numbers.

Drag these milestones into the order in which they happened, earliest at the top.

Cardano gets 5 ± √−15 for "sum 10, product 40" and calls it useless
Mahavira writes that a negative number has no square root
Euler introduces the symbol i for √−1
Bhaskara II repeats the warning in his Bijaganita
Hamilton defines a + ib as an ordered pair (a, b)
Girard accepts square roots of negative numbers

Nearly a thousand years from "impossible" to "an ordered pair". In this chapter you will meet that new number, learn to add, multiply and divide with it, and give every complex number a home as a point of a plane.

Sina