Ex 2 - Multiply complex numbers

- TO MULTIPLY THE GIVEN COMPLEX NUMBERS WE'LL MULTIPLY THESE ALGEBRAICALLY LIKE WE NORMALLY WOULD THEN WE NEED TO MAKE SURE THE PRODUCT IS IN THE FORM OF A COMPLEX NUMBER MEANING IT'S IN THE FORM OF A + OR - B1 WHERE "A" IS A REAL NUMBER AND + OR - B1 IS AN IMAGINARY NUMBER WHERE "I" IS EQUAL TO THE SQUARE ROOT OF -1. SO FOR 5I x -6I SO IT'S 5 x -6 IS -30 AND "I" x "I" WOULD BE EQUAL TO "I" SQUARED. THIS WOULD BE OUR PRODUCT BUT IT'S NOT IN THE CORRECT FORM BECAUSE IT'S NOT IN THE FORM OF A COMPLEX NUMBER. BUT SINCE WE KNOW THAT "I" IS EQUAL TO THE SQUARE ROOT OF -1, AND "I" SQUARED = -1, WE CAN SUBSTITUTE -1 FOR "I" SQUARED. SO WE'D HAVE -30 x -1 = 30. NEXT WE HAVE -4I x THE QUANTITY, 3 - 7I. SO HERE WE'LL BE DISTRIBUTING 4I, WELL 4I x 3 WOULD BE 12I. THEN WE HAVE 4I x -7I, THAT'S GOING TO BE -28I SQUARED OR -28I SQUARED. AGAIN THIS IS OUR PRODUCT BUT IT'S NOT IN THE CORRECT FORM OR THE FORM OF A COMPLEX NUMBER. WE CAN'T HAVE "I" SQUARED IN OUR FINAL PRODUCT BUT BECAUSE "I" SQUARED = -1, WE'LL PERFORM A SUBSTITUTION FOR "I" SQUARED. WE WOULD HAVE 12I - 28 x -1. NOTICE HOW THIS WOULD BECOME 28 AND NORMALLY WHEN WRITING A COMPLEX NUMBER, WE WANT TO PUT THE REAL PART FIRST SO WE'LL WRITE THIS AS 28 AND THIS AS A 12I, SO WE'LL HAVE 12I. NEXT WE HAVE 3I CUBED OR 3I RAISED TO THE THIRD POWER. SO WE CAN APPLY OUR RULES FOR EXPONENTS OR WE CAN JUST RECOGNIZE THIS AS 3I x 3I x 3I AND SINCE 3 x 3 x 3 = 27, AND "I" x "I" x "I" = "I" CUBED, THIS WOULD BE OUR PRODUCT BUT AGAIN IT'S NOT IN THE CORRECT FORM BECAUSE IT'S NOT IN THE FORM OF A COMPLEX NUMBER. BUT WE DO KNOW THAT "I" SQUARED = -1. SO TO SIMPLIFY THIS WE COULD WRITE THIS AS 27 x "I" SQUARED x "I" THIS IS STILL "I" TO THE THIRD, BUT NOW WE CAN SUBSTITUTE -1 FOR "I" SQUARED. WE'D HAVE 27 x -1 x "I" SO WE HAVE -27I. AGAIN FOR THESE PRODUCTS, WE CAN'T HAVE "I" RAISED TO A POWER HIGHER THAN 1. AND WE CAN ALSO VERIFY THESE PRODUCTS ON THE GRAPHING CALCULATOR. LET'S MAKE SURE WE'RE IN COMPLEX MODE SO I'LL PRESS THE MODE KEY AND HIGHLIGHT "A" + DI DOWN HERE. SO ONCE IT'S HIGHLIGHTED WE PRESS ENTER, GO BACK TO THE HOME SCREEN BY PRESSING SECOND MODE, THEN WE JUST TYPE IN OUR PRODUCTS. SO WE HAVE 5I, THE "I" IS SECOND DECIMAL POINT x -6I WHICH VERIFIES OUR FIRST PRODUCT. NEXT WE HAVE 4I x THE QUANTITY 3 - 7I. THIS VERIFIES OUR SECOND PRODUCT AND THEN FOR OUR THIRD EXAMPLE, WE HAVE 3I RAISED TO THE THIRD. HERE'S THE EXPONENT KEY AND THIS IS RAISED TO THE THIRD POWER, WHICH VERIFIES OUR THIRD PRODUCT. OKAY, I HOPE YOU FOUND THIS HELPFUL.