3.2 Complex Numbers, Real and Imaginary Parts

If \(b\) is a real number, a number of the form \(bi\) is called an imaginary number; \(5i\), \(-12i\), \(17i\) and \(-102i\) are examples. If \(a\) and \(b\) are both real, a number of the form \(a+bi\) is called a complex number, as are \(3+2i\), \(7-23i\) and \(-14+35i\). The set of all complex numbers is denoted by \(\mathbb {C}\).

Every real number \(a\) is the complex number \(a+0i\), so \(\mathbb {R}\) sits inside \(\mathbb {C}\) — which is what the figure below records.

Let \(z=x+iy\) be a complex number. Then \(x\) is called the real part of \(z\) and \(y\) the imaginary part, written \[\operatorname {Re}z=x,\qquad \operatorname {Im}z=y .\] Note that the imaginary part is the real number \(y\), not \(iy\).

Let \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\) be two complex numbers. Then

1.

\(z_1=z_2\) if and only if \(x_1=x_2\) and \(y_1=y_2\).

2.

\(z_1\pm z_2 = (x_1\pm x_2)+i(y_1\pm y_2)\)

For example, \[(4+3i)+(7-5i)=(4+7)+i(3-5)=11-2i .\]

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