Computation in Positional Systems

Transcription

1 Survey of Math - MAT 40 Page: Computation in Positional Systems Addition To operate in other Bases, unlike the book, I think that it is easier to do the calculations in Base 0, and then convert (using division, remainder method) to whatever Base we are working in. For instance, if we are in Base and we are adding 4 C 3 I would add the numbers in Base 0 and convert this to Base, So: 4 C 3 D 7 7 D r 4 C 3 D This idea of making the remainder after division the next place value is a simplistic version of how we converted our numbers into other bases. Example Using the basic rules of adding and the idea of converting into base after the calculation, we should be able to add two large numbers. 3 4 C 3 4 Step : Add C 4 in base C 4 D D r D So must be split, with the quotient going to the next place value and the remainder going down as part of our answer:

2 Survey of Math - MAT 40 Page: Step : Add C 4 C 3 in Base C 4 C 3 D D r3 D 3 So one goes up to next place value and 3 gets dropped as part of our answer: Step 3: Add C 3 C in Base C 3 C D D r D So one goes to the next place value (down in this case) and the other becomes part of the answer: Which is the solution to the problem.. Adding in "unusual" Bases: Example What about A 3 C E 9 Here we must remember that A=0, C=, and E=4

3 Survey of Math - MAT 40 Page: 3 Step : C C 9 D C 9 D D r D high, low A 3 C E 9 Step : C 3 C E D C 3 C 4 D D r D high and low A 3 C E 9 Step 3: C A C D C 0 C D D r0 D 0 left and 0 low A 3 C E 9 0 Subtraction When subtracting a larger value from a smaller place value in our system we borrow 0 for the next place down. When we are in a different base, we borrow whatever Base Value that we are in. Remember that we cannot subtract a larger number from a smaller one.

4 Survey of Math - MAT 40 Page: 4 Example Step : We cannot do 3; so we will have to borrow from the next place value.3/ and add to the ones place. Then subtract what is there: = Step : We cannot do 4; so we will borrow from the and add to the./ Subtract what is left = = Example 4 Subtract in Base T E Step : Since we can't do 9 E; we will have to borrow from the./ and add to the 9 4 9= T E T Notice here that E D D 0; and that 0 D T Step : Again we cannot do 4 9; so we will have to borrow another one from the T and

5 Survey of Math - MAT 40 Page: add to the 4: The subtract, just remember that T D 0 and E D : 4= 9 4 9= T E 7 T 3 Multiplication Again, I would practice doing multiplication of a 3-digit number.39/ and a -digit number.4/ to get a feel for what we are about to do. We will do all the calculations in Base 0, and then convert to the other bases after. Example 3 4 Step : Multiply 4 3 in base = = r 0 Step : Multiply 4 in base = 0 0 = r 4 Step 3: Multiply 4 in base = = 3 r 3

6 Survey of Math - MAT 40 Page: Example Two digit multiplication: Step : 4 4 = 0 0 = r 4 Step : 4 3 = 7 7 = r Step 3: = = 4 r Next Line using (remember to move over a place value to the left!) Step 4: = = r 0

7 Survey of Math - MAT 40 Page: 7 Step : = 7 7 = 0 r 7 Step : = 4 4 = r Step 7: The nal answer is begotten by adding the two rows in Base, remember any number or larger has to be converted: For example: C 7 D D r

8 Survey of Math - MAT 40 Page: 4 Division Practice your long division techniques with some Base 0 numbers. You will nd that to do Base Division that you will have to rst come up with the Divisors Multiplication Facts. Example 7 Find 7 /4 7 To do this we will rst have to get the multiplication facts of in base 7: Then the division is easy, remember that we are in Base 7 and to subtract appropriately if we have to borrow. Example If the number we are dividing has more that one place value, we will have to be careful coming up with the Mult. Facts. /379F To nd the Mult Facts, use Column Form Multiplication: x x 4 x 3 3 x 4 4 x A x C x 7 7E x 90 x 9 A x A B4 x B C x C D x D EA x E FC x F 0E Then the problem becomes easy, as long as you remember that we are in Base and that A D 0; B D ; C D ; D D 3; E D 4; F D

9 Survey of Math - MAT 40 Page: 9 37 fi379f 3 9 7F 7E So: /379F D 37 with a remainder of

10 Survey of Math - MAT 40 Page: 0 Exercises Exercise Add 47 and 3 Exercise Add CE and 7B Exercise 3 Evaluate: Exercise 4 Evaluate: B4 47A Exercise Find the product of: 3 and 4 Exercise Find the product of: 43 and 3 Exercise 7 Evaluate: Exercise Evaluate: 404 4

11 Survey of Math - MAT 40 Page: Solutions Solution Add 47 and 3 47 C 3 D 3 Solution Add CE and 7B CE C 7B D 4 Solution 3 Evaluate: D Solution 4 Evaluate: B4 47A B4 47A D 3 Solution Find the product of: 3 and D 4 Solution Find the product of: 43 and D 3 Solution 7 Evaluate: D 3 7 Solution Evaluate: D 3