09-18-2009Examples of Addition in Non-decimal BasesExample 1: Compute (313)four+ (22)fourin base four.I suggest you do this by way of a place value diagram. Set the problem up as follows42s 41s 40s3 1 3+ 2 2Notice this is the same as how I make a place value diagram for regular addition except instead of100s, 10s and 1s, I use 42s, 41s, and 40s as my place values.Now add the columns up. If any number in your answer is bigger or equal to the base four put it inparentheses.42s 41s 40s3 1 3+ 2 23 3 (5)The numbers in parenthesis are not proper digits in base four so I must make some exchanges. Sincethe (5) in the 40column is not a proper base four number, I must rewrite it. But 5 is 1 four and 1 one.So I add 1 in the 41s column and replace the (5) in the ones column with a 1. This gives me42s 41s 40s3 1 3+ 2 23 3 (5)3 (4) 1Now the (4) in the 41s column is not a proper base four number so I must make an exchange. Thedigit 4 is one 4 and 0 ones so I add 1 to the 42column and replace (4) with 0. This gives me42s 41s 40s3 1 3+ 2 23 3 (5)3 (4) 1(4) 0 1Now (4) is not a proper base four number so I must rewrite it for my final answer. The digit 4 is 1four and 0 ones. So I insert a column for 43s and add 1 to it then replace the (4) with a 0. This givesme42s 41s 40s3 1 3+ 2 23 (3) (5)3 (4) 1(4) 0 11 0 0 1So my answer is(313)four+ (22)four= (1001)four.Example 2: Compute (320)four− (221)fourin base four.I suggest you do this by way of a place value diagram. Set the problem up as follows42s 41s 40s3 2 0− 2 2 1Notice this is the same as how I make a place value diagram for regular subtraction except insteadof 100s, 10s and 1s, I use 42s, 41s, and 40s as my place values.Now we want to do the subtraction starting from the 40s column, however we can’t take 1 from 0 sowe must exchange one of the 41s for 4 40s. We get:42s 41s 40s3 2 0− 2 2 13 1 (4)− 2 2 1But now since we can’t take 2 from 1 in the 41s column we’ll have to make another exchange. Tradeone 42for four 41s. We get42s 41s 40s3 2 0− 2 2 13 1 (4)− 2 2 12 (5) (4)− 2 2 1Now we can subtract column by column. Doing this we get:42s 41s 40s3 2 0− 2 2 13 1 (4)− 2 2 12 (5) (4)− 2 2 10 3 3So our answer is(320)four− (221)four=
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