Math, imaginary Curves, lines   Math   Trigonometry   Calculus   Area & Volume ```i⁻³ = i¹ = i⁵ = i⁹ = +i i⁻² = i² = i⁶ = i¹⁰ = –1 i⁻¹ = i³ = i⁷ = i¹¹ = –i i⁰ = i⁴ = i⁸ = i¹² = +1 i^(4n) = 1 (n is integer) i^(4n + 1) = i i^(4n + 2) = -1 i^(4n + 3) = -i √i = ±((√2/2)(1+i) √(x+iy) = √[ (r+x)/2 ] ± i √[ (r–x)/2 ] with r = √(x² + y²) ∛i = –i, ½+√3/2, –½+√3/2 ∛(–1) = –1, ½–i√3/2, and ½–i√3/2 ∛(–N) = ∛N[ –1, ½–i√3/2, and ½–i√3/2 ] de Moivre's formula, states that for any complex number (and, in particular, for any real number) x and integer n it holds that" (cos x + i sin x)ⁿ = cos (nx) + i sin (nx) i = √(–1) Euler's formula is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x: e^(ix) = cos x + i sin x Euler's Identity e^(iπ) + 1 = 0 i = √(–1) e is Euler's number, the base of natural logarithms π is the ratio of the circumference of a circle to its diameter. Derive square root of i One way to obtain this answer is to solve the equation (a+bi)=√i for a and b, or (a+bi)²=i If you expand this equation using the rules for complex multiplication, you get (a²–b²) + (2ab)i = 0+i. Equating real and imaginary parts gives you a²–b²=0 and 2ab=1. The equation a²–b²=0 means a=±b. However, if you plug a=–b into the second equation you get –2b²=1 which can not be satisfied by any real number b. Therefore, the case a=–b is not possible, meaning a must equal b. Then the second equation becomes 2a²=1. This means either a=b=1/√2 or a=b=–1/√2. That gets you the two square roots of i: (1/√2)(1+i) and (–1/√2)(1+i) Cube root ∛1 = 1, –(1/2) + i√3/2, –(1/2) – i√3/2 ``` Home Area, Volume Atomic Mass Black Body Radiation Boolean Algebra Calculus Capacitor Center of Mass Carnot Cycle Charge Chemistry   Elements   Reactions Circuits Complex numbers Constants Curves, lines deciBell Density Electronics Elements Flow in fluids Fourier's Law Gases Gravitation Greek Alphabet Horizon Distance Interest Magnetics Math   Trig Math, complex Maxwell's Eq's Motion Newton's Laws Octal/Hex Codes Orbital Mechanics Particles Parts, Analog IC   Digital IC   Discrete Pendulum Planets Pressure Prime Numbers Questions Radiation Refraction Relativistic Motion Resistance, Resistivity Rotation Series SI (metric) prefixes Skin Effect Specific Heat Springs Stellar magnitude Thermal Thermal Conductivity Thermal Expansion Thermodynamics Trigonometry Units, Conversions Vectors Volume, Area Water Wave Motion Wire, Cu   Al   metric Young's Modulus