Calculatrice de nombres complexes
Effectuer des opérations sur des nombres complexes étape par étape
La calculatrice tentera de simplifier n'importe quelle expression complexe, avec des étapes indiquées. Elle effectue des additions, des soustractions, des multiplications, des divisions, des élévations à la puissance, et trouve la forme polaire, le conjugué, le module et l'inverse du nombre complexe.
Solution
Your input: simplify and calculate different forms of (1+3i)(5+i)
Use FOIL to multiply (for steps, see foil calculator), don't forget that i2=−1:
((1+3i)(5+i))=(2+16i)
Hence, (1+3i)(5+i)=2+16i
Polar form
For a complex number a+bi, polar form is given by r(cos(θ)+isin(θ)), where r=√a2+b2 and θ=atan(ba)
We have that a=2 and b=16
Thus, r=√(2)2+(16)2=2√65
Also, θ=atan(162)=atan(8)
Therefore, 2+16i=2√65(cos(atan(8))+isin(atan(8)))
Inverse
The inverse of 2+16i is 12+16i
In general case, multiply the expression 1a+ib by the conjugate (the conjugate of a+ib is a−ib):
1a+ib=1(a−ib)(a+ib)(a−ib)
Expand the denominator: 1(a−ib)(a+ib)(a−ib)=a−iba2+b2
Split:
a−iba2+b2=aa2+b2−iba2+b2
In our case, a=2 and b=16
Therefore, (12+16i)=(1130−4i65)
Hence, 12+16i=1130−4i65
Conjugate
The conjugate of a+ib is a−ib: the conjugate of 2+16i is 2−16i
Modulus
The modulus of a+ib is √a2+b2: the modulus of 2+16i is 2√65
Answer
(1+3i)(5+i)=2+16i=2.0+16.0i
The polar form of 2+16i is 2√65(cos(atan(8))+isin(atan(8)))
The inverse of 2+16i is 12+16i=1130−4i65≈0.00769230769230769−0.0615384615384615i
The conjugate of 2+16i is 2−16i=2.0−16.0i
The modulus of 2+16i is 2√65≈16.1245154965971