- WELCOME TO A SECOND LESSON ON CONVERTING BASE 10 NUMBERS
TO DIFFERENT BASES.
IN THIS METHOD WE'LL BE USING A CALCULATOR
TO CONVERT TO DIFFERENT BASES.
LET'S BEGIN BY CONSIDERING THE NUMBER 1432 IN DIFFERENT BASES.
IN BASE 10, THE BASE WE'RE USED TO WORKING WITH,
WE KNOW THE ONE REPRESENTS 1, 1000.
WE KNOW THE FOUR REPRESENTS FOUR 100'S.
WE KNOW THE THREE REPRESENTS 3 10'S
AND THE TWO REPRESENTS 2 1'S
GIVING US OUR NUMBER IN BASE 10 OF 1,432.
ANOTHER WAY TO REPRESENT THE PLACE VALUES IN BASE 10
WOULD BE TO TAKE OUR BASE 10 AND START WITH 10 TO THE ZERO
WHICH WOULD GIVE US THE ONES AND THEN MOVE TO THE LEFT
BY ADDING ONE TO THE EXPONENT EACH TIME.
TEN TO THE FIRST WOULD BE THE 10'S PLACE VALUE,
10 TO THE SECOND WOULD BE THE HUNDREDS,
10 TO THE THIRD WOULD BE THE THOUSANDS AND SO ON.
SO IF WE LOOK AT THIS SAME NUMBER IN A DIFFERENT BASE,
THE PLACE VALUES CHANGE.
FOR BASE FIVE,
NOTICE HOW WE START WITH FIVE TO THE ZERO WITH THE ONES,
BUT THEN WE'D HAVE FIVE TO THE FIRST WITH THE FIVES
FOLLOWED BY FIVE TO THE SECOND OR THE 25'S PLACE VALUE.
THEN FIVE TO THE THIRDS OR 100 25'S PLACE VALUE AND SO ON.
IF WE CHANGE TO BASE SIX,
NOTICE HOW AGAIN THE PLACE VALUES CHANGE.
AND SO IF WE WANTED TO WRITE THESE NUMBERS
IN BASE FIVE AND SIX,
IN BASE 10 THE ONE NOW REPRESENTS ONE, 125,
THE FOUR REPRESENTS FOUR 25'S.
THE THREE REPRESENTS THREE FIVES AND THE TWO REPRESENTS TWO ONES
GIVING US 242 IN BASE 10.
FOR A 1432 BASE SIX
NOTICE HOW THIS WOULD BE EQUAL TO 380 IN BASE 10.
BUT THE MAIN GOAL IN THIS LESSON
IS TO START WITH A NUMBER IN BASE 10
AND WRITE IT USING A DIFFERENT BASE.
TO DO THIS, WE'LL BE USING A CALCULATOR METHOD
WHERE NUMBER ONE WILL FIND THE HIGHEST POWER OF THE BASE B
THAT WILL DIVIDE INTO THE GIVEN NUMBER AT LEAST ONCE
AND THEN DIVIDE.
TWO, WE'LL KEEP THE WHOLE NUMBER PART
AND MULTIPLY THE FRACTIONAL OR DECIMAL PART BY BASE B.
STEP THREE, WE'LL REPEAT STEP TWO,
KEEPING THE WHOLE NUMBER PART INCLUDING ZERO'S,
AND CARRY THE FRACTIONAL OR DECIMAL PART TO THE NEXT STEP
UNTIL A WHOLE NUMBER RESULT IS OBTAINED.
STEP FOUR WE'LL COLLECT ALL THE WHOLE NUMBER PARTS
TO GET THE NUMBER IN BASE B.
BEFORE WE TAKE A LOOK AT OUR TWO EXAMPLES, THOUGH,
LET'S SEE WHY THIS METHOD WORKS.
IF WE CONSIDER THE NUMBER 2,573 IN BASE 10,
NOTICE HOW THE HIGHEST POWER OF 10
THAT WOULD DIVIDE AT LEAST ONCE INTO THE GIVEN NUMBER
WOULD BE 10 TO THE THIRD OR 1,000.
SO IF WE TAKE 2,573 AND DIVIDE IT BY 1,000, IT GIVES US 2.573.
SO OF COURSE THE TWO TELLS US
THAT WE HAVE TWO THOUSANDS IN 2,573,
BUT NOTICE HOW THE DECIMAL PART
REPRESENTS THE FRACTIONAL PART OF A THOUSAND
THAT REMAINS AFTER DIVIDING BY 1,000.
SO WE TAKE JUST A DECIMAL PART
AND MULTIPLY IT BY THE BASE OF 10.
NOTICE HOW THIS GIVES US 5.73
WHERE THE FIVE TELLS US HOW MANY HUNDREDS REMAIN.
THIS FIVE REPRESENTS 500.
TAKE THIS DECIMAL PART MULTIPLY IT BY THE BASE 10.
IT TELLS US HOW MANY 10S REMAIN.
THERE ARE SEVEN 10'S OR 70 AND THEN FINALLY THE POINT 3 x 10
GIVES US HOW MANY ONES REMAIN, THERE ARE THREE ONES.
NOW LET'S TAKE A LOOK AT OUR EXAMPLES.
WE WANT TO CONVERT THE BASE 10 NUMBER, 214, TO BASE THREE.
I THINK IT'S ALWAYS HELPFUL TO WRITE OUT THE PLACE VALUES
FOR THE GIVEN BASE WHICH I'VE ALREADY DONE HERE BELOW,
THREE TO THE ZERO, ONE'S PLACE VALUE,
THREE TO THE ONE WOULD BE THE THREES PLACE VALUE,
THREE TO THE SECOND WOULD BE THE NINES PLACE VALUE AND SO ON.
SO TO BEGIN, WE WANT TO FIND THE HIGHEST POWER OF THREE
THAT WE DIVIDE AT LEAST ONCE INTO 214.
WELL NOTICE THREE TO THE FIFTH WOULD BE TOO LARGE
SO WE'LL DIVIDE BY THREE TO THE FOURTH OR 81.
BECAUSE WE'LL BE DIVIDING BY THREE TO THE FOURTH,
NOTICE HOW OUR NUMBER IN BASE THREE
IS GOING TO HAVE ONE, TWO, THREE, FOUR, FIVE PLACE VALUES.
LET'S GO AHEAD AND SET THAT UP.
ONE, TWO, THREE, FOUR, FIVE
JUST SO WE DON'T MISS ANY PLACE VALUES.
SO NOW WE'RE GOING TO TAKE 214
AND DIVIDE BY THREE TO THE FOURTH OR 81.
NOW WE'LL TAKE THE WHOLE NUMBER PART OR TWO,
WHICH WILL BE THE DIGIT IN THE 81'S PLACE VALUE.
SO WE'LL PUT A TWO HERE, AND NOW WE'RE GOING TO SUBTRACT THE TWO
WHICH WILL LEAVE THE DECIMAL PART SO - 2, ENTER,
AND NOW EACH TIME I'M GOING TO MULTIPLY BY THE BASE
WHICH IN THIS CASE WOULD BE THREE.
SO x 3 ENTER, NOW WE'RE GOING TO TAKE THE WHOLE NUMBER PART AGAIN
OF THE ONE WHICH WILL GIVE US THE DIGIT INTO THE THREE
TO THE THIRD PLACE VALUE OR 27'S PLACE VALUE.
SO WE'LL PUT A ONE HERE
AND NOW WE'LL SUBTRACT THE WHOLE NUMBER PART OR SUBTRACT ONE,
ENTER, AND THEN MULTIPLY IT BY THE BASE AGAIN.
OUR BASE IS THREE, SO x 3 ENTER.
TAKE THE WHOLE NUMBER PART OF TWO,
THAT'S OUR NEXT DIGIT FOR THE THREE TO THE SECOND
OR NINES PLACE VALUE.
SUBTRACT THE WHOLE NUMBER OF TWO AND MULTIPLY IT BY THREE AGAIN.
SO x 3, ENTER.
THE NEXT DIGIT WILL BE TWO
WHICH WILL BE IN THE THREE TO THE FIRST OR THREES PLACE VALUE.
SUBTRACT THE WHOLE NUMBER OF TWO AND MULTIPLY IT BY THREE AGAIN.
NOTICE HOW THE RESULT IS A WHOLE NUMBER SO WE KNOW WE'RE DONE.
THIS IS THE DIGIT IN THE ONE'S PLACE VALUE.
SO WHAT WE'VE DISCOVERED IS THAT 214 IN BASE 10
IS EQUAL TO TWO, ONE, TWO, TWO, ONE IN BASE THREE.
TO CHECK THIS, LET'S GO AHEAD AND EXPAND TWO, ONE, TWO, TWO,
ONE IN BASE THREE.
THIS TWO REPRESENTS 2 x 3 TO THE FOURTH.
THIS ONE REPRESENTS 1 x 3 TO THE THIRD.
THIS TWO REPRESENTS 2 x 3 TO THE SECOND.
THIS TWO REPRESENTS 2 x 3 TO THE FIRST
AND FINALLY THIS ONE REPRESENTS 1 x 3 TO THE ZERO.
WE SIMPLIFY WE DO GET 214 BASE 10 VERIFYING OUR ANSWER.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
WE'RE GOING TO CONVERT THE BASE 10 NUMBER OF 13,967
TO BASE SEVEN.
NOTICE TO BEGIN I WROTE OUT THE PLACE VALUES FOR BASE SEVEN.
WE'RE LOOKING FOR THE HIGHEST POWER OF SEVEN
THAT WE DIVIDE AT LEAST ONCE INTO 13,967
WHICH WOULD BE SEVEN TO THE FOURTH.
SEVEN TO THE FIFTH WOULD BE TOO LARGE.
SO BECAUSE WE'RE DIVIDING BY SEVEN TO THE FOURTH
TO BEGIN, OUR NUMBER IN BASE SEVEN
IS GOING TO HAVE ONE, TWO, THREE, FOUR, FIVE PLACE VALUES
OR FIVE DIGITS, SO ONE, TWO, THREE, FOUR, FIVE.
AND NOW WE'LL DIVIDE.
13,967 DIVIDED BY SEVEN TO THE FOURTH OR 2,401 GIVES US THIS.
WE TAKE ONLY THE WHOLE NUMBER PART FOR OUR FIRST DIGIT
WHICH IS FIVE.
AND NOW WE'LL SUBTRACT THE WHOLE NUMBER PART,
THEN MULTIPLY BY BASE SEVEN.
SO - 5 AND THEN x 7,
THE NEXT DIGIT IS FIVE THE WHOLE NUMBER PART.
AND NOW WE'LL SUBTRACT FIVE AND MULTIPLY BY SEVEN AGAIN.
THE NEXT DIGIT IS ALSO A FIVE.
SUBTRACT THE WHOLE NUMBER PART AND MULTIPLY BY SEVEN.
NOTICE HERE THERE'S NO WHOLE NUMBER PART
AND THEREFORE THE NEXT DIGIT WOULD BE ZERO.
AND WE MULTIPLY BY SEVEN AGAIN.
THIS SHOULD BE THE LAST DIGIT.
SO THIS PRODUCT SHOULD BE A WHOLE NUMBER WHICH IT IS.
OUR LAST DIGIT IS TWO.
SO WE HAVE FOUND THAT 13,967 BASE 10
IS EQUAL TO FIVE, FIVE, FIVE, ZERO, TWO BASE SEVEN.
AND AGAIN JUST TO CHECK THIS, LET'S GO AHEAD AND EXPAND
FIVE, FIVE, FIVE, ZERO, TWO IN BASE SEVEN
WHICH I'VE ALREADY DONE HERE
JUST TO VERIFY THAT OUR ANSWER IS CORRECT.
I HOPE YOU FOUND THIS HELPFUL.