How do you add, subtract, multiply, and divide the complex numbers z = −10 + 1 i and w = 7 i?

1 Answer
Feb 19, 2017

Answer:

# z+w = -10 + 8i#

# z-w = -10 -6i#

# \ \ \ \ \ zw = -7-70i #

# \ \ \ \ \ z/w = 1/7+10/7i#

Explanation:

We have:

# z=-10+i#, and #w=7i#

To add and subtract complex numbers we simply add subtract the real and imaginary parts separately, thus:

# z+w = (-10+i) + (7i)#
# " "= (-10) + (1+7)i#
# " "= -10 + 8i#

And:

# z-w = (-10+i) - (7i)#
# " "= (-10) + (1-7)i#
# " "= -10 -6i#

To multiply complex numbers we multiply every combination in one term with every combination of the other term, and use #i^2=-1#, so

#zw = (-10+i)(7i) #
# " "= (-10)(7i)+(i)(7i) #
# " "= -70i+7i^2 #
# " "= -70i+7(-1) #
# " "= -7-70i #

And for division we generally remove the complex denominator by multiplying the numerator and denominator by the complex conjugate of the denominator (as the product of a complex number with its conjugate is always real). As the denominator in this example is purely imaginary we can multiply the numerator and denominator by #i# to make it real.

#z/w = (-10+i)/(7i) #
# " " = (-10+i)/(7i) *i/i#
# " " = (-10i+i^2)/(7i^2)#
# " " = (-10i-1)/(-7)#
# " " = -(1+10i)/(-7)#
# " " = 1/7+10/7i#