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Welcome to NRICH.

Complex Numbers

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Basics

• 3 i²=-1
• 4 Article: An Introduction to Complex Numbers
• 5 Is i less than or greater than 1?
• 5 Square root of i
• 5 Square root of a complex number
• 5 Squared Root of -1
• 5 Practical Uses of Imaginary Numbers
• 5 Uses of Complex Numbers
• 5 Solving quadratic with complex coefficients
• 5 Solving complex quadratics
• 5 Complex Numbers, leading to some university level algebra
• 5 z/(z-i) problem
• 5 |z₁ + z₂| £ |z₁| + |z₂|
• 5 |z-1|=1
• 5 Squared Root of -1
• 6 No Order in Complex Numbers
• 5 Solving az+bz^*=c

Euler's formula and De Moivre's theorem

[That is, e^ix = cos x + i sin x, and (cos x + i sin x)ⁿ = cos nx + i sin nx.]

• 5 What is e^z?
• 5 e^ix = cos x + isin x and related concepts
• 5 Euler's Formula
• 5 e^ix = cos x + i sin x
• 5 Sin(pi/10) as a surd expression
• 5 Complex roots
• 5 Converting complex numbers expressed in different forms
• 5 e^ip=-1
• 5 De Moivre's Theorem: does it apply only to rationals?
• 5 Series for trigonometric functions and e^x
• 5 Taking roots to find extra values of e^ip?
• 6 Physics applications of e^ip=-1

Locus

• 5 Complex transformations and loci
• 5 Locus of arg(z/(z-1+2i)) = p/3
• 5 Locus of z such that |z-15j| = 4|z+30|

Complex numbers and geometry

• 5 1/z transformation
• 5 Nice regular polygon theorem
• 6 Graphing Complex Solutions
• 6 Domains
• 6 Roots of polynomial in an annulus

Miscellaneous

• 5 cosq+isinq does not equal 0
• 5 Proof Of the Newton Raphson method for finding complex roots
• 5 Complex roots question
• 5 Sqrt(ab) = sqrt(a)sqrt(b)
• 5 i ⁱ
• 5 Need conceptual help with -1⁰...
• 5 Hyperbolic functions, natural logs and complex numbers
• 5 3/2 - 9/20 + 15/56 - 21/110 + .....
• 5 Graphs of imaginary functions in MathCAD
• 5 Quaternions
• 5 A Proof that 2pi=0?
• 6 Convergent series of complex numbers
• 6 Is this a surjection?
• 6 Complex Analysis Discussion
• 6 Analytic continuation