MATH1131 Chap.3 Complex Numbers and the Argand Diagram
Complex Numbers and the Argand Diagram
Complex numbers extend the real numbers so that every polynomial equation has solutions. A complex number a + bi has a real part and an imaginary part, both real numbers, and arithmetic follows the usual rules with i squared replaced by -1. Division works by multiplying the numerator and denominator by the conjugate of the denominator, which makes the denominator a positive real number.
The Argand diagram plots a + bi at the point (a, b), so the modulus becomes a distance, addition becomes vector addition, conjugation becomes reflection in the real axis, and multiplication by i becomes a quarter-turn. That geometric view turns many questions into pictures: sets described by equations or inequalities in |z - w| are circles, discs or perpendicular bisectors.
The chapter also covers square roots of complex numbers found by matching real and imaginary parts, a technique that relies on the extra modulus equation to reduce the work to simple linear steps.
What this chapter covers
- 01
Real and imaginary parts
- 02
Arithmetic and division by the conjugate
- 03
Properties of the complex conjugate
- 04
The Argand diagram and modulus
- 05
Distance between complex numbers
- 06
Square roots by equating parts
- 07
Sketching regions defined by moduli
Worked example · free
Simplifying a quotient and checking its modulus
- 1Multiply top and bottom by 2 + 3i: (5 + i)(2 + 3i) / (4 + 9) = (10 + 15i + 2i + 3i^2)/13.
- 1Simplify with i^2 = -1: (7 + 17i)/13, so the answer is 7/13 + (17/13)i.
- 1Check: |5 + i|/|2 - 3i| = sqrt(26)/sqrt(13) = sqrt(2), and |7 + 17i|/13 = sqrt(338)/13 = sqrt(2).
Key terms
- Imaginary part
- The real coefficient b in a + bi, not the term bi itself.
- Complex conjugate
- The number a - bi obtained by changing the sign of the imaginary part of a + bi.
- Modulus
- The distance of a complex number from the origin in the Argand diagram, equal to the square root of a squared plus b squared.
- Argand diagram
- The plane in which each complex number a + bi is plotted as the point (a, b).
Complex Numbers and the Argand Diagram FAQ
Why multiply by the conjugate when dividing complex numbers?
A number times its conjugate is a squared plus b squared, a positive real number. Multiplying numerator and denominator by the denominator's conjugate therefore leaves an ordinary real denominator and an answer in the form a + bi.
Is the imaginary part of 3 + 4i equal to 4i?
No. By definition the imaginary part is the real number 4. Writing 4i is a common slip that causes errors when parts are compared in equations.
How do I find the square roots of a complex number by hand?
Set (a + bi) squared equal to the number, equate real and imaginary parts, and add the modulus equation a squared plus b squared equals the modulus of the number. Solve for a squared and b squared, then match signs using the imaginary part.
What shape is |z - 2i| < 3 in the Argand diagram?
It is the open disc of radius 3 centred at 2i, that is, at the point (0, 2). The boundary circle is excluded because the inequality is strict, so draw it dashed.
How can I tell that a complex expression is real?
A complex number is real exactly when it equals its own conjugate, equivalently when its imaginary part is zero. Taking the conjugate of an expression is often faster than expanding it.
Can complex numbers be compared with less than or greater than?
No. Complex numbers have no ordering compatible with arithmetic, so inequalities only make sense for real quantities such as moduli, real parts or imaginary parts. Sketching questions therefore always place inequalities on one of those real quantities.
Exam move
Alternate between algebraic and geometric versions of the same question. For each arithmetic exercise, plot the inputs and the result on an Argand diagram and see whether the picture agrees, for example whether a product has the expected length. For sketching questions, substitute z = x + iy, square moduli to remove square roots, and simplify to a familiar Cartesian curve before drawing.
Keep a list of the conjugate properties and practise using them to shorten proofs, since showing that an expression equals its conjugate is a frequent exam step. For each region you sketch, test one point inside and one outside against the original condition; this habit catches reversed inequalities and wrong centres before they cost marks.
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