- WHEN DIVIDING COMPLEX NUMBERS
OR WRITING COMPLEX NUMBERS IN FRACTION FORM,
WE'RE NOT ALLOWED TO HAVE "I" IN THE DENOMINATOR.
REMEMBER "I" IS EQUAL TO THE SQUARE ROOT OF -1,
SO THIS RULE IS VERY SIMILAR TO NOT HAVING A SQUARE ROOT
IN THE DENOMINATOR OF A RADICAL EXPRESSION.
SO WHEN THE DENOMINATOR IS JUST AN IMAGINARY NUMBER,
WE COULD ELIMINATE THE "I" FROM THE DENOMINATOR
BY USING THE FACT THAT "I" SQUARED IS EQUAL TO -1.
NOTICE IF WE MULTIPLY THIS BY I/I,
THE DENOMINATOR WOULD BE 3I SQUARED
AND THEN WE COULD SUBSTITUTE - 1 FOR "I" SQUARED,
THIS WOULD SIMPLIFY TO -3.
BUT WE DO PREFER THE DENOMINATOR TO BE POSITIVE.
SO INSTEAD OF MULTIPLYING BY I/I,
WE'RE ACTUALLY GOING TO MULTIPLY BY -I/-I.
AND NOW, WE'LL GO AHEAD AND MULTIPLY.
IN THE NUMERATOR, WE'LL HAVE -6I,
AND THEN WE'LL HAVE + 7I SQUARED,
AND NOTICE THE DENOMINATOR IS GOING TO BE -3I SQUARED.
AND NOW, WE'RE GOING TO SUBSTITUTE -1 FOR "I" SQUARED
BOTH IN THE DENOMINATOR AND THE NUMERATOR.
SO WE'RE GOING TO HAVE -6I + 7 x -1 ALL/-3 x -1.
AND NOW, LET'S SIMPLIFY THIS AGAIN.
I'M GOING TO GO AHEAD AND PUT THE REAL PART FIRST
AND THE IMAGINARY PART SECOND IN THE NUMERATOR.
SO NOTICE HOW WE'D HAVE -7 - 6I,
AND WE CAN LEAVE THE QUOTIENT IN THIS FORM,
BUT SOMETIMES THIS IS ALSO BROKE UP INTO 2 TERMS
WHERE WE HAVE THE REAL NUMBER FIRST
AND IMAGINARY NUMBER SECOND.
SO IF WE DID THAT, NOTICE HOW WE'D HAVE -7/3 - 6/3I.
BUT 6/3 SIMPLIFIES TO 2, SO WE HAVE -7/3 - 2I.
IT CAN ALSO BE EXPRESSED IN THIS FORM.
NOW, LET'S LOOK AT OUR SECOND EXAMPLE.
WHERE WE HAVE A COMPLEX NUMBER A DENOMINATOR
THAT HAS BOTH A REAL PART AND AN IMAGINARY PART,
WE'RE GOING TO MULTIPLY BOTH THE NUMERATOR AND DENOMINATOR
BY THE COMPLEX CONJUGATE OF THE DENOMINATOR.
SO BECAUSE THE DENOMINATOR IS -3 + 4I,
WE'RE GOING TO MULTIPLY BY -3 - 4I/-3 - 4I.
SO WE'LL HAVE 4 PRODUCTS IN THE NUMERATOR
AND 4 PRODUCTS IN THE DENOMINATOR.
SO WE'LL HAVE -2 x -3 THAT'S 6, -2 x -4I THAT'S +8I,
+5I x -3I THAT'S -15I,
AND THEN +5I x -4I THAT'S -20I SQUARED.
AND THEN THE DENOMINATOR WILL HAVE -3 x -3 THAT'S 9,
AND THEN -3 x -4I THAT'S +12I,
THEN WE HAVE +4I x -3I THAT'S -12I,
AND THE LAST PRODUCT IS GOING TO BE 4I x -4I
OR -16I SQUARED.
SO NOW, WE'RE GOING TO COMBINE THE MEASURING NUMBERS
AND SUBSTITUTE -1 FOR "I" SQUARED.
SO IN THE NUMERATOR, WE'RE GOING TO HAVE 6,
AND THEN WE HAVE 8 - 15I THAT'S -7I
OR - 7I - 20 x -1 FOR "I" SQUARED.
NOTICE IN THE DENOMINATOR,
WE HAVE OPPOSITE MEASURING NUMBERS HERE AND HERE,
THAT'S 0, SO WE HAVE 9 - 16,
AND THEN AGAIN "I" SQUARED IS EQUAL TO -1.
SO LOOKING AT THE NUMERATOR, WE'RE GOING TO HAVE 6,
SO THE +20, SO 26 - 7I, THE DENOMINATOR IS GOING TO BE 9
AND THIS WILL BE +16 OR 25.
SO WE CAN EXPRESS THIS QUOTIENT IN THIS FORM
OR AGAIN, IF WE WANT TO,
WE CAN BREAK THIS UP INTO 2 TERMS.
THE REAL PART WOULD BE 26/25
AND THE MEASURING PART WOULD BE -7/25I.
OKAY. I HOPE YOU FOUND THIS HELPFUL.