1. Chapter 7 Section 7.2 Addition & Subtraction in Different Bases

2. Adding Numbers in Different Bases Adding numbers in different bases requires the need to have learned the basic addition facts in another base. The table below give the basic addition facts for base four. +0404 1414 2424 3434 0404 0404 1414 2424 3434 1414 1414 2424 3434 10 4 2424 2424 3434 11 4 3434 3434 10 4 11 4 12 4 The reasoning for how we have gotten some of the entries is shown below. 2 4 + 2 4 = 4 (base 10) = 10 4 2 4 + 3 4 = 5 (base 10) = 11 4 3 4 + 3 4 = 6 (base 10) = 12 4 Below is shown how the standard addition algorithm is applied to solve addition problems in base four. 1203434 +1332424 1414 023 111 2312424 +202424 0404 213 11 99 +126 225 Converting to base 10 182 +34 216

3. Addition of Numbers Using the Lattice Method Another way to organize the addition of numbers is to use the lattice method. It works similar to how you use it with multiplication but you fill in the addition facts in the correct columns. The first problem shows how to use this in base 10 to add 849+5767 and the second shows how it is used in base 4 to add. 849 +5767 1 6 1 0 1 55 0 2312424 +202424 1 0 0 1 1 12 0 6166 3120404 Try the following addition problems in the given bases. You have to figure out the basic addition facts as you are doing the problems. 2413535 +1342525 1 0 1 0 1 23 0 4310505 4052626 +5323636 0 5 1 1 0 33 1 3415656 1

4. 10100111 2 + 110110 2 2 1 Adding & Subtracting Binary (Base 2) Numbers Addition and subtraction of base 2 numbers can be accomplished by know how to do the problems below and following the rules for carrying and borrowing just like in base 10. 0 + 0 0 1 + 0 1 0 + 1 1 + 1 10 1 0 + 0 1 + 0 10 1 0 + 1 10 1 + 1 11 No Carrying digit. Carried a 1 into the next digit. Add each of the following base 2 numbers. 101101 2 + 10110 2 2 110 1 0 0 01 11 1 0 1 0 1 1 1 1 1 1

5. Subtraction also has the same idea as that of base 10 (decimal). In subtraction we will either need to borrow or we won't. Here are the basic subtraction problems. 0 - 0 0 1 - 0 1 - 1 0 No Borrowing needed. 10 - 1 1 100 - 1 11 Carried a 1 into the next digit. 1 1 1 1 Here are the steps needed to do the subtraction problem below. 101011 2 - 10101 2 2 1. Subtract 1's digits. 2. Subtract 2's digits. 3. Borrow from 8's digit to 4's digit. 4. Subtract 4's digits. 5. Subtract 8's digits. 6. Borrow from 32's digit to 16's digit. 7. Subtract 16's digits. 1 0 1 1 1 1 0