Week Five.

Tuesday February 5

Today in math class we worked through some problems which were not in the common base 10 form. Instead of base 10, we used base 4, 6, 9... ect.

Some examples of these change of base problems are like so:

Time Problem:

 3 hrs 15 min
-2 hrs 45 min
 0 hrs 30 min = 12:30pm

The way that we got that answer was starting with the minutes first. 5 min - 5 min is obviously 0 min. Then we have to subtract 40 min from 10 min, since we dont have enough minutes anymore, we have move over to the hours place and borrow one hour. Since we are borrowing one hour and moving it to the minutes place, we give 60 min. So then we subtract 70-40 which is 30. Then we do the remaining 2 hrs-2 hrs. And end up with a time of 0 hrs 30 min. but since the hours are a base 12, the 0 hrs actually is supposed to be a 12. So we get a final time of 12:30.

Some simpiler problems are like so...

  32
+23  (Base 4)
 121

In a base 4 system, no numbers can ever be above a 4 ever, once a number reaches 4, it has to move to the next largest place value slot. So we start with the normal 2+3 which is 5, but since we are in a base 4 system fives dont exist, so we move a group of for to the tens place, and keep 1 in the ones place. Then we add up the 3+2 again, but we also have an extra 1 from the ones place that moved over. So we get 6, which is actually a group of 4 and two left over, so we move the group of four over to the hundreds place, and keep the 2 in the tens place. This results in a final answer of 121.

Some more problems are like so:


  345
+266 (Base 7)
  644

  43
+25 (Base 6)
 112

  333
+101 (Base 4)
 1100

Subtraction problems with a change of base use the same rules as normal subtraction, it just takes a little more effort and thinking...

 63                              
-25 (Base 7)
 35

 We start with the ones and realize that we need to borrow a group of ones from the next place value over, but since this problem is in base seven, one group in the tens place is equivalent to seven individuals in the once place. So the 60 becomes a 50, and the 3 becomes a 10. This is because we  borrowed 7 from the tens, and gave it to the ones, and 3+7=10. So then we move on with our subtraction problem with 10-5 which is 5. Then we move on over to the tens place and subtract 5-2 because we borrowed a group from the tens. We end up with an answer of 35.

Some more problems like this:

 31
-12 (Base Four)
 13

 342
-253 (Base Six)
   45

 302
-103 (Base Four)
 133

The next step of this process is to be given a problem, and then try to determine what base the problem is in:

  231
+414
1200

So in this problem like most others you start from the right and work your way left. So we start with the 4+1. Since in this problem the 4+1 results in a 0 in the ones place, we can guess that this problem is in a base 5. We can assume this because there is a 0 in the ones place, and if this happens, a group of five moves to the tens place. Then you move to the tens place, and then do the 3+1+1 (the second one is being carried from the ones place). So we get an answer of 5 again, so we carry one over to the hundreds place. Then we do 4+2+1 which equals 7, but we are in base ten, so we move one group of five over to the thousands place, and keep 2 in the hundreds. Resulting in an answer of 1200.

More problems like this:

  231
+414
1045

This problem is a base six because all the numbers are below a base 6 in the problem, and 400+200=600 which in this case would equal 1000 because its a base 6 problem.

  344
+143
1042

This problem is in a base 5.

Subtraction Problems like this are done the exact same way:

 523
-254
 236

This problem is in the base of 7. Since we cant subtract 4 from three we have to borrow a group from the tens place. but we dont know how many we have to borrow since we dont know what base this problem is in, but we do know that whatever number we borrow, has to result in a subtraction problem that equals 6. so we ask ourselves, what plus three - 4 is six? The answer is 7. 7+3=10. 10-4=6. So from here we can assume the problem is in base 7, and finish the problem to double check and make sure.

More problems:

 523
-254                        (Base 8)

 427


1020
-203                         (Base 5)
 312