- WELCOME TO A LESSON ON COMPLEX NUMBERS.
THE GOALS OF THIS VIDEO ARE TO IDENTIFY
IMAGINARY AND COMPLEX NUMBERS
AND ALSO TO SIMPLIFY IMAGINARY AND COMPLEX NUMBERS.
SO IF YOU WERE ASKED TO SOLVE THE EQUATION X SQUARED + 1 = 0,
YOU MAY TRY TO FACTOR THIS.
BUT THAT WOULDN'T WORK,
BECAUSE IT'S NOT A DIFFERENCE OF SQUARES.
THEN YOU MAY RESORT TO THE QUADRATIC FORMULA,
AND THAT WOULDN'T WORK.
BUT WE COULD ALSO JUST SOLVE THIS EQUATION
USING SQUARE ROOTS,
MEANING IF WE ISOLATE THE X SQUARED BY SUBTRACTING 1
ON BOTH SIDES, WE WOULD HAVE X SQUARED = -1.
NOW, WE CAN TAKE THE SQUARE ROOT OF BOTH SIDES,
AND WE COME UP WITH A SOLUTION THAT SAYS
X = + OR - THE SQUARE ROOT OF -1,
OR X EQUALS POSITIVE SQUARE ROOT -1,
OR X EQUALS NEGATIVE SQUARE ROOT -1.
THIS IS ACTUALLY A NEW TYPE OF SOLUTION
WHICH LEADS TO A NEW SET OF NUMBERS,
BECAUSE WE'RE NOT USED TO HAVING
THE SQUARE ROOT OF NEGATIVE NUMBERS.
BUT IN FACT IF WE SUB THESE INTO THE ORIGINAL EQUATION FOR X,
REMEMBER IF YOU SQUARE A SQUARE ROOT, THEY UNDO EACH OTHER,
SO THESE ARE SOLUTIONS TO THIS EQUATION.
THIS LED TO A NEW SET OF NUMBERS
AND TWO NEW DEFINITIONS
THAT "I" IS EQUAL TO THE SQUARE ROOT OF -1
AND "I" SQUARED IS EQUAL TO -1.
SO AGAIN, THIS LED TO A NEW SET OF NUMBERS
CALLED THE IMAGINARY NUMBERS
WHICH THEN LED TO THE SET OF COMPLEX NUMBERS.
FIRST, IMAGINARY NUMBERS ARE THOSE
IN THE FORM OF THE SQUARE ROOT OF "A" WHERE "A"
IS A NEGATIVE REAL NUMBER.
HERE ARE THREE EXAMPLES OF IMAGINARY NUMBERS.
WE CAN SIMPLIFY THESE VERY SIMILAR
TO HOW WE SIMPLIFY SQUARE ROOTS
EXCEPT NOW WE HAVE A NEW DEFINITION
TO DEAL WITH THE NEGATIVE.
SO WE COULD REWRITE THIS AS THE SQUARE ROOT OF -1 x 64
WHICH IS EQUAL TO THE SQUARE ROOT OF -1 x THE SQUARE ROOT
OF 64.
WELL NOW BY DEFINITION, THIS IS EQUAL TO "I,"
THIS IS EQUAL TO 8,
SO THIS SIMPLIFIES TO THE IMAGINARY NUMBER 8I.
ON THIS NEXT ONE, THIS IS -2 x THE SQUARE ROOT -16 IS -2
x THE SQUARE ROOT OF -1 x 16.
SO REALLY IT'S VERY SIMILAR TO WHAT WE'VE BEEN DOING
IN THE PAST WITH SQUARE ROOTS,
WE JUST HAVE AN EXTRA FACTOR OF -1 WHICH GIVES US AN "I."
SO YOU CAN PROBABLY SEE THIS WOULD BE -2 TIMES,
THIS WOULD BE EQUAL TO "I," THIS WOULD BE EQUAL TO 4,
SO WE'D HAVE -2 x 4I WHICH IS EQUAL TO -8I.
THE SQUARE ROOT OF -27 IS EQUAL TO -1 x 27.
27 IS NOT A PERFECT SQUARE, SO IT WOULD BE 3 x 3 x 3.
SO THE SQUARE ROOT OF -1 SIMPLIFIES TO "I,"
THE SQUARE ROOT OF 9 SIMPLIFIES TO 3,
SO THIS WOULD GIVE US A 3I,
AND THEN WE'RE LEFT WITH SQUARE ROOT 3.
NOW IF WE TRIED TO SOLVE THE EQUATION
X SQUARED - 10X + 34 = 0,
THAT ALSO DOESN'T FACTOR SO WE WOULD APPLY
THE QUADRATIC FORMULA,
AND I'VE ALREADY DONE THIS TO SAVE SOME TIME.
BUT WHAT'S NEW ABOUT THIS IS
WHEN WE SIMPLIFY THE DISCRIMINANT,
WE END UP GETTING THE SQUARE ROOT OF -36,
WHICH LEADS TO PART OF THIS ANSWER BEING IMAGINARY.
SO WE HAVE 10 + OR - 6I, ALL DIVIDED BY 2.
WE DIVIDE BOTH TERMS AND NUMERATOR BY 2,
WE HAVE 5 + 3I.
SO THERE IS AN IMAGINARY NUMBER AS PART OF THIS SOLUTION.
BUT THERE'S ALSO A REAL PART
AND THAT LED TO ANOTHER SET OF NUMBERS
CALLED THE COMPLEX NUMBERS.
COMPLEX NUMBERS ARE THOSE NUMBERS
WRITTEN IN THE FORM OF "A" + BI WHERE "A" AND B ARE REAL NUMBERS
AND "I" IS THE SQUARE ROOT OF -1.
SO WE HAVE A REAL PART, THAT'S THE "A" PART,
AND WE HAVE THE IMAGINARY PART, THE BI PART,
AND "A" + BI IS CALLED THE STANDARD FORM
OF A COMPLEX NUMBER.
SO IF WE WANTED TO SIMPLIFY THESE COMPLEX NUMBERS
IN THE FORM OF "A" + BI,
WE STILL NEED TO SIMPLIFY THE IMAGINARY PART.
SO WE'D HAVE 7 - SQUARE ROOT OF -1 x 4.
WE KNOW NOW THIS WILL GIVE US "I,"
WE KNOW THE SQUARE ROOT OF 4 IS 2,
SO THIS WOULD SIMPLIFY TO 7 - 2I.
AND AGAIN, IT'S IN THE FORM OF "A" + BI.
LET'S TAKE A LOOK AT ONE MORE HERE.
AFTER YOU DO THESE AWHILE,
YOU PROBABLY LOOK AT THAT AND SEE 3I.
BUT FOR THE PURPOSE OF SHOWING WORK,
LET'S GO AHEAD AND REWRITE THIS AS -1 x 9/4,
SO WE'D HAVE 6 PLUS,
NOW THE SQUARE ROOT OF -1 WILL GIVE US "I",
THE SQUARE ROOT OF 9 WILL GIVE US 3,
THIS IS EQUAL TO 3I x 4.
THAT WOULD BE 12I ALL DIVIDED BY 4.
NOW, BE CAREFUL SIMPLIFYING HERE.
WE CANNOT JUST SIMPLIFY THE 6 AND THE 4.
WE CANNOT SIMPLIFY ACROSS ADDITION.
SO WE SHOULD REWRITE THIS AS TWO SEPARATE FRACTIONS,
6 DIVIDED BY 4 + 12I DIVIDED BY 4,
THIS GIVES US 3 HALVES + 3I, AGAIN IN THE FORM OF A + BI.
LET'S TAKE A LOOK AT HOW THE SETS OF NUMBERS
RELATE TO ONE ANOTHER WITH A VENN DIAGRAM.
HERE WE HAVE THE NATURAL NUMBERS, WHOLE NUMBERS,
INTEGERS, RATIONAL, IRRATIONAL NUMBERS, REAL NUMBERS,
THEN THE NEW SETS, THE COMPLEX NUMBERS
AND THE IMAGINARY NUMBERS.
AND NOW THAT WE KNOW THAT COMPLEX NUMBERS
HAVE BOTH A REAL PART AND AN IMAGINARY PART,
YOU CAN SEE THAT THIS IS GOING
TO BE THE MOST ENCOMPASSING SET OF NUMBERS.
SO THE COMPLEX NUMBERS WOULD BE OUT HERE
IN THE FORM OF "A" + BI.
NEXT, SINCE THE COMPLEX NUMBERS ARE MADE UP
OF A REAL PART AND IMAGINARY PART, THESE TWO SETS
WOULD BE THE REAL NUMBERS AND THE IMAGINARY NUMBERS.
SO THE REAL NUMBERS, DENOTED WITH A CAPITAL R,
WOULD BE HERE AND THE IMAGINARY NUMBER WOULD BE OVER HERE.
THEY DO NOT OVERLAP.
NOW, THE REMAINING SETS ARE ONES WE'VE ALREADY BEEN WORKING WITH
THROUGHOUT OUR STUDY OF ALGEBRA,
SO LET'S GO AHEAD AND START WITH THIS INNERMOST RECTANGLE.
IF YOU TAKE A LOOK AT THE REMAINING SETS HERE,
WE HAVE NATURAL NUMBERS, WHOLE NUMBERS, INTEGERS,
RATIONAL AND IRRATIONALS.
WELL, THE NATURAL NUMBERS ARE 1, 2, 3, OR THE COUNTING NUMBERS.
THEY WOULD BE THE INNERMOST SET.
NEXT, THE ONLY DIFFERENCE
BETWEEN WHOLE NUMBERS AND NATURAL NUMBERS
IS THAT THE SET OF WHOLE NUMBERS INCLUDES ZERO.
SO THAT WOULD BE THE NEXT SET.
NEXT COMES THE INTEGERS.
INTEGERS CONSIST OF POSITIVE AND NEGATIVE
NATURAL NUMBERS AND ZERO, SO THAT WOULD BE THE NEXT SET.
AND NOW, WE'RE LEFT WITH TWO SETS
AND THAT'S GOING TO CONSIST
OF THE RATIONAL AND IRRATIONAL NUMBERS.
NATURAL NUMBERS, WHOLE NUMBERS AND INTEGERS ARE RATIONAL,
SO THEY WOULD BE WITHIN THE SET OF RATIONAL NUMBERS
WHICH LEAVES THIS SET OVER HERE TO BE THE IRRATIONAL NUMBERS.
SO IT IS IMPORTANT TO KNOW HOW THE SETS OF NUMBERS
RELATE TO ONE ANOTHER,
SO YOU MIGHT CONSIDER SKETCHING ONE OF THESE DIAGRAMS
INTO YOUR NOTES.
WE WILL COVER THE OPERATIONS OF COMPLEX NUMBERS
IN THE NEXT VIDEO.
THANK YOU FOR WATCHING.