- WELCOME TO A LESSON ON SCIENTIFIC NOTATION.
THE GOALS OF THIS VIDEO ARE TO CONVERT
BETWEEN DECIMAL NOTATION, AND SCIENTIFIC NOTATION,
AND ALSO TO MULTIPLY AND DIVIDE NUMBERS
IN SCIENTIFIC NOTATION.
SCIENTIFIC NOTATION, ALSO KNOWN AS EXPONENTIAL NOTATION
PROVIDES A USEFUL WAY OF WRITING
VERY LARGE OR VERY SMALL NUMBERS.
FOR EXAMPLE, THE U.S. NATIONAL DEBT
IS NOW OVER $12 TRILLION 900 BILLION,
WHICH COULD BE WRITTEN IN SCIENTIFIC NOTATION
AS 1.9 x 10 TO THE THIRTEENTH POWER.
THE TIME IT TAKES LIGHT TO TRAVEL 1 METER
IN DECIMAL NOTATION IS EQUAL TO THIS,
WHICH CAN BE WRITTEN IN SCIENTIFIC NOTATION AS
3 x 10 TO THE -9 POWER.
SCIENTIFIC NOTATION FOR A NUMBER IS IN THE FORM OF
A x 10 TO THE POWER OF M.
WHERE A IS A NUMBER IN DECIMAL FORM
THAT IS AT LEAST ONE BUT LESS THAN TEN.
AND M, THE POWER ON TEN, MUST BE AN INTEGER.
SO IF YOU TAKE A LOOK AT THIS POWER OF 10,
IF M IS POSITIVE,
YOU CAN SEE IT WILL BE MULTIPLYING
BY NUMBERS THAT ARE LARGER THAN 1.
THEREFORE, WHEN M IS POSITIVE
AND WE CONVERT TO DECIMAL NOTATION,
THE NUMBER WILL BE LARGER THAN THE VALUE OF A.
IF WE'RE MULTIPLYING BY A NEGATIVE POWER OF 10,
WE WOULD MULTIPLY A BY SOME FRACTION.
AND IF WE MULTIPLY A BY A NUMBER THAT'S LESS THAN ONE
THE RESULT WILL BE A NUMBER THAT IS LESS THAN A.
NOW, THAT'S IMPORTANT BECAUSE WHEN WE CONVERT
FROM SCIENTIFIC NOTATION TO DECIMAL NOTATION,
WE WANT TO UNDERSTAND
WHY WE'RE MOVING THE DECIMAL POINT EITHER LEFT OR RIGHT.
SO IF M IS POSITIVE
THE DECIMAL POINT WILL MOVE TO THE RIGHT M PLACES
IN DECIMAL NOTATION, MAKING THE NUMBER LARGER THAN A.
IF M IS NEGATIVE THE DECIMAL POINT WILL MOVE LEFT,
THE OPPOSITE VALUE OF M PLACES,
BECAUSE THE DECIMAL NOTATION WILL BE LESS THAN A.
SO WE LOOK AT THIS FIRST EXAMPLE,
IF WE'RE MULTIPLYING BY A POSITIVE POWER OF 10,
IN DECIMAL NOTATION THIS NUMBER WILL BE LARGER THAN A.
AND, IN FACT, TO FIND THAT NUMBER,
WE HAVE TO MOVE THE DECIMAL POINT RIGHT 10 PLACES.
LET'S GO AHEAD AND GIVE IT A TRY.
WE START WITH THE VALUE OF A,
AND WE HAVE TO MOVE RIGHT TEN PLACES.
AFTER MOVING THE DECIMAL THREE PLACES WE END UP HERE,
AND WE HAVE TO MOVE IT SEVEN MORE PLACES TO THE RIGHT.
THEREFORE, WE ARE GOING TO HAVE TO ADD
SEVEN ZEROS TO THIS NUMBER.
SO THERE'S 3, 4, 5, 6, 7, 8, 9, 10.
NOW, TO MAKE IT CLEAR WHAT OUR NUMBER IS,
WE SHOULD REWRITE THIS.
SO 1.245 x 10 TO THE TENTH POWER IS = TO 12 BILLION 450 MILLION.
NOW, ON THE SECOND EXAMPLE WE CAN SEE WE'RE MULTIPLYING
-8.75 x 10 TO THE POWER OF -8.
MULTIPLYING BY A NEGATIVE POWER OF TEN
WILL MAKE THE NUMBER IN DECIMAL FORM LESS THAN -8.75.
SO NOW WE HAVE TO MOVE THE DECIMAL PLACE --
SO NOW WE HAVE TO MOVE THE DECIMAL POINT LEFT EIGHT PLACES.
LET'S GO AHEAD AND GIVE IT A TRY.
NOW, THIS NEGATIVE SIGN IS KIND OF IN THE WAY,
SO I'M GOING TO ELIMINATE IT TEMPORARILY.
NOW WE'LL START MOVING LEFT EIGHT PLACES.
SOON AS WE MOVE IT ONCE THERE'S NO MORE DIGITS,
SO WE HAVE TO ADD, AGAIN, ANOTHER SEVEN ZEROS.
SO THERE'S 1, 2, 3, 4, 5, 6, 7, 8 PLACES TO THE LEFT.
SO OUR NUMBER IN DECIMAL NOTATION,
REMEMBER IT IS A NEGATIVE VALUE,
SO WE HAVE -875 TEN BILLIONTHS.
NEXT WE'LL CONVERT A NUMBER IN DECIMAL NOTATION
TO SCIENTIFIC NOTATION.
AND THE STEPS ARE MOVE THE DECIMAL TO FORM A,
SUCH THAT A IS AT LEAST ONE BUT LESS THAN TEN.
THE NUMBER OF PLACES THE DECIMAL MOVES
WILL INDICATE THE VALUE OF M.
IF YOU MOVE TO THE LEFT M IS POSITIVE,
IF YOU MOVE TO THE RIGHT M IS NEGATIVE.
AGAIN, WE DON'T WANT TO MEMORIZE THIS,
WE WANT TO MAKE SENSE OF IT.
SO THE FIRST STEP IS TO DETERMINE
WHERE THE DECIMAL SHOULD BE
SO THAT A IS AT LEAST ONE BUT LESS THAN TEN,
AND THAT WOULD BE HERE.
SO FAR WE KNOW THAT IN SCIENTIFIC NOTATION THIS WILL BE
7.6 x SOME POWER OF 10.
THE QUESTION IS HOW MANY PLACES DID THEY MOVE IT TO GET IT HERE?
WELL, IT STARTED HERE,
SO IF WE COUNT OVER TO WHERE WE WANT IT TO BE
WE'D HAVE 1, 2, 3, 4, 5, 6, 7 PLACES.
IT WOULD BE A POSITIVE SEVEN.
SINCE THIS IS A RATHER LARGE NUMBER,
IT MAKES SENSE THIS X1,
IT WOULD HAVE TO BE POSITIVE.
ON OUR SECOND EXAMPLE,
WE KNOW TO FORM A NUMBER THAT IS AT LEAST ONE BUT LESS THAN TEN,
THE DECIMAL WOULD HAVE TO BE HERE,
SO THE VALUE OF OUR A WOULD BE 3.76.
OKAY. THIS IS A VERY SMALL NUMBER,
WHICH INDICATES THIS WILL BE A NEGATIVE EXPONENT.
BUT LET'S COUNT THE NUMBER OF PLACES
WE HAD TO MOVE IT TO THE RIGHT, 1, 2, 3, 4, 5, 6.
THEREFORE, OUR EXPONENT IS -6.
OKAY. NOW LET'S TALK ABOUT
HOW YOU CAN MULTIPLY AND DIVIDE NUMBERS IN SCIENTIFIC NOTATION.
IF THESE TWO NUMBERS ARE IN SCIENTIFIC NOTATION,
EVERYTHING HERE IS BEING MULTIPLIED TOGETHER.
SO IT'S PRETTY STRAIGHT FORWARD,
WE SIMPLY MULTIPLY A x B,
THEN MULTIPLY 10 TO THE M x 10 TO THE N.
AND TO DIVIDE NUMBERS IN SCIENTIFIC NOTATION
WE DIVIDE A BY B,
AND DIVIDE 10 TO THE M POWER BY 10 TO THE N POWER.
NOW, ONE WORD OF CAUTION,
A x B AND A DIVIDED BY B MUST BE AT LEAST ONE
BUT LESS THAN TEN
TO BE IN THE CORRECT SCIENTIFIC NOTATION FORM.
SO WE MAY HAVE TO DO A MINOR ADJUSTMENT
IF A x B OR A DIVIDED BY B DOES NOT FIT THIS REQUIREMENT.
OKAY. SO LET'S GO AHEAD AND MULTIPLY THESE TOGETHER.
SO THE FIRST STEP IS TO SET IT UP.
THIS WOULD BE 7.1 x 1.3 x 10 TO THE THIRD x 10 TO THE FOURTH.
SO WE'RE REALLY NOT DOING MUCH
OTHER THAN CHANGING THE ORDER OF MULTIPLICATION.
TO SAVE SOME TIME,
I'VE ALREADY PERFORMED THIS MULTIPLICATION
9.23 x 10 TO THE POWER OF 3 + 4 WOULD GIVE US 10 TO THE SEVENTH.
NOW, A IS AT LEAST ONE BUT LESS THAN TEN,
SO WE LEAVE OUR ANSWER IN THIS FORM.
ON THIS NEXT EXAMPLE WE SET IT UP THE SAME WAY,
BUT WHEN WE FIND THIS PRODUCT TO GET
72.15 x 10 TO THE POWER OF +3, THIS IS THE CORRECT VALUE,
BUT NOT IN THE CORRECT FORM.
REMEMBER A HAS TO BE LESS THAN TEN,
SO WE NEED TO CONVERT THIS TO SCIENTIFIC NOTATION.
THIS WOULD BE THE SAME AS
7.215 x 10 TO THE FIRST x 10 TO THE THIRD.
SO NOW THE VALUE OF A IS AT LEAST ONE BUT LESS THAN TEN.
SO NOW WE JUST HAVE TO
MULTIPLY 10 TO THE FIRST x 10 TO THE THIRD
TO GET OUR FINAL ANSWER,
WHICH WOULD BE x 10 TO THE FOURTH.
CHECK THIS ON THE GRAPHING CALCULATOR,
WE CAN OPEN THE SET OF PARENTHESIS, TYPE IN 9.25.
IF WE PRESS SECOND COMMA
THIS CAPITAL E STANDS FOR SCIENTIFIC NOTATION.
THEN WE TYPE IN THE X POINTED ON THE 10,
AND THEN SAME FOR THE SECOND 7.8,
SECOND COMMA, AND THEN -3.
CLOSE THE PARENTHESIS, HIT ENTER.
NOW THE ANSWER'S IN DECIMAL FORM, BUT WE CAN SEE
THAT IF WE CONVERT THIS TO SCIENTIFIC NOTATION
IT WOULD MATCH OUR ANSWER.
LET'S GO AHEAD AND TRY A COUPLE DIVISION PROBLEMS.
SO WE BREAK THIS INTO TWO PARTS.
THE FIRST PART WILL BE
8.4 DIVIDED BY 6 x 10 TO THE TWELFTH
DIVIDED BY 10 TO THE FOURTH.
8.4 DIVIDED BY 6 = 1.4.
10 TO THE TWELFTH DIVIDED BY 10 TO THE FOURTH,
WE SUBTRACT OUR EXPONENTS 10 TO THE EIGHTH.
EVERYTHING IS IN THE CORRECT FORM AND WE'RE DONE.
TO SAVE SOME TIME ON THIS LAST PROBLEM
I'M JUST GOING TO GO AHEAD AND THINK OF THIS
AS TWO DIVISION PROBLEMS.
1.2 DIVIDED BY 6.87
IS APPROXIMATELY 0.179 X 10 TO THE NEGATIVE TWELFTH
DIVIDED BY 10 TO THE -3,
-12 - -3 WOULD BE 10 TO THE NEGATIVE NINTH.
AGAIN, THIS IS NOT IN THE CORRECT FORM
FOR SCIENTIFIC NOTATION,
BUT WE CAN REWRITE THIS AS 1.79 x 10 TO THE POWER OF -1
SINCE WE MOVED IT TO THE RIGHT ONE PLACE,
AND THEN OF COURSE x 10 TO THE NEGATIVE NINTH.
SO OUR FINAL ANSWER WILL BE
APPROXIMATELY 1.79 x 10 TO THE POWER OF -10.
LET'S GO AHEAD AND CHECK THIS ON THE GRAPHING CALCULATOR.
1.23, SECOND COMMA, -12 DIVIDED BY 6.87, SECOND COMMA, -3.
AND YOU CAN SEE OUR APPROXIMATION IS CORRECT.
I HOPE THIS LESSON WAS HELPFUL.
THANK YOU FOR WATCHING.