Complex Numbers and Quadratic Equations
Class 11 Mathematics · Chapter 4
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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: