Complex numbers:

When you add the real number and an imaginary number, like 4+ j7 or 45- j83, you obtain a complex number. This term does not mean complicated; it would better be called as composite.  But again, original name stuck, even if it was not the best possible thing. Real numbers are 1 dimensional. They can be depicted on the line. Imaginary numbers are one dimensional for the same reason. But complex numbers need 2 dimensions to be defined completely.

Adding and subtracting complex numbers

Adding complex numbers is just the matter of adding the real parts and the complex parts separately. The sum of 4+j7 and 45 - j83 is thus (4+45)+ j(7-83)=49+ j( -76)=49- j76. BY subtracting the complex numbers works similarly. The difference (4+ j7)-(45- j83) is found by multiplying the 2nd complex number by -1 and then adding the result, getting (4+ j7)+(-1(45 - j83) = (4 + j7) + ( -45+ j83)= -41+ j90.

The general formula for sum of 2 complex numbers (a+jb) and (c+jd) is

(a+ jb) + (c + jd) = (a + c)+ j(b+d)

The plus and minus number signs becomes tricky when working with sums and differ- ences of complex numbers. Just remember that any difference  can be treated as a sum: multiply the second number by   1 and then add. You may be wanting to do some exercises to get yourself acquainted with way these numbers behave, but in the working with the engineers, you will not be called upon to deal with the complex numbers at level of nitty-gritty.

If you want to become an engineer, you will need to practice adding and subtracting the complex numbers. But it is not difficult once you get used to it by solving a few sample problems.

Multiplying complex numbers

You should know that how complex numbers can be multiplied, to have a full understanding of the behavior. When you multiply these numbers, you need to treat them as the sums of number pairs, as binomials.

It is easier to give general formula than to work with the specifics here. The product of (a + jb) and (c  + jd) is equal to ac+ jad+ jbc+jjbd. While simplifying, remember that jj =-1, so you obtain the final formula:

(a + jb)(c + jd) = (ac - bd) +j(ad +bc)

With the addition and subtraction of the complex numbers, you should be careful with the signs (plus and minus). And with the addition and subtraction, you get used to doing these problems with the little practice. Engineers sometimes have to multiply the complex numbers.