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The multiplication of complex numbers in cartesian form is binomial multiplication

function ccal()fh=document.forms<0>;aa=fh.a.value;bb=fh.b.value;cc=fh.c.value;dd=fh.d.value;aam=aa*cc-bb*dd;fh.am.value=aam;bbm=aa*dd-(-1)*bb*cc;fh.bm.value=bbm;cd2=cc*cc-(-1)*dd*dd;aad=(aa*cc-(-1)*bb*dd)/cd2;fh.ad.value=aad;bbd=(bb*cc-aa*dd)/cd2;fh.bd.value=bbd

For a + jb = + j

and c + jd = + j

then (a + jb)(c + jd) = + j

Division is a bit an ext involved in cartesian type and requires the process called rationalization the the facility number.


(a + jb)/(c + jd) = + j

IndexComplex numbers
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Go BackRationalization of complex Numbers

The department of facility numbers which space expressed in cartesian type is promoted by a procedure called rationalization. The development of a fraction


presents difficulties since of the imaginary component of the denominator. The denominator can be forced to be genuine by multiply both numerator and denominator by the conjugate the the denominator.


Expanding puts the an outcome of the department in cartesian form again.

Calculation for multiplication and divisionIndexComplex numbers
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Go BackComplex Conjugate

The conjugate that a complex number is that number through the authorize of the imaginary part reversed


The energy of the conjugate is the any complicated number multiplied by its complicated conjugate is a actual number:


This operation has practical utility for the rationalization of complex numbers and the square root of the number times its conjugate is the magnitude of the complex number once expressed in polar form.

ApplicationsIndexComplex numbers
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Go BackApplications of complex Conjugates

When a real positive definite quantity is required from a actual function, the square the the role can it is in used. In the case of a complicated function, the facility conjugate is offered to achieve that purpose. The product that a complex number and its complicated conjugate is the complicated number analog to squaring a genuine function. The complicated conjugate is offered in the rationalization of complex numbers and also for finding the amplitude of the polar form of a complex number.

One applications of the complicated conjugate in physics is in recognize the probability in quantum mechanics.

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Since the wavefunction which specifies the probability amplitude may be a complicated function, the probability is identified in regards to the complicated conjugate to attain a actual value.