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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 1 4 3 2 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 hundreds we know the three represents three tens and the 2 represents two ones giving us our number in base 10 of 1000 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 1 to the exponent each time 10 to the first would be the tens 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 to a different base the place value is change for base five notice how we'd start with five to the zero with the ones but then we'd have five to the first of the five followed by five to the second or the 25 place value then five to the thirds or 125 is place value and so on if we change the base six notice how again the place value is change so if we wanted to write these numbers in base five and six in base 10 the one now represents one 125 the full represents for 25 the 3 represents three fives that she represents two ones giving us 242 in base 10 4 1 4 3 2 base 6 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 2 will keep the whole number part and multiply the fractional or decimal part by be step three we'll repeat step two keeping the whole number part including zeros and carry the fractional or decimal part to the next step until a whole number result is obtained step four will 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 2573 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 1000 so if we take 2573 and divide by 1000 it gives us two point five seven three so of course the 2 tells us that we have two thousands in 2573 but notice how the decimal part represents the fractional part of a thousand that remains after dividing by 1000 so if we take just a decimal part and multiply by the base of 10 notice how this gives us five point seven 3 where the 5 tells us how many hundreds remain this 5 represents five hundred take this decimal part multiplied by the base 10 it tells us how many tens remain there are seven tens or seventy and then finally the point 3 times 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 3 I think it's always helpful to write out the place values for the given base which I've already done here below 3 to the 0 the ones place value 3 to the 1 would be the 3 place value 3 to the 2nd would be the 9s place value and so on so to begin we want to find the highest power of 3 then we'd divide at least once into 214 well notice 3 to the 5th would be too large so we'll divide by 3 to the 4th or 81 because we'll be dividing by 3 to the 4th notice how our number in base 3 is going to have 1 2 3 4 5 place values let's go ahead and set that 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 80 ones 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 minus 2 enter and now each time we're going to multiply by the base which in this case will be 3 so x 3 enter now we're going to take the whole number part again to the 1 which will give us the digit into the 3 to the third place value or 27s place value so we'll put a 1 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 3 so x 3 enter take the whole number part of 2 that's our next digit for the 3 to the 2nd or 9s place value subtract the whole number of 2 and multiply by 3 again times 3 enter the next digit will be 2 which will be in the 3 of the first or 3s place value subtract the whole number of 2 and multiply by 3 again notice how the result is a whole number so we know we're done this is the digit in the ones place value so what we've discovered is that 214 in

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