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Section 4.4

Math in Our World. Section 4.4. Operations in Base Number Systems. Learning Objectives. Add in bases other than 10. Subtract in bases other than 10. Multiply in bases other than 10. Divide in bases other than 10. Adding in other Bases.

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Section 4.4

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  1. Math in Our World Section 4.4 Operations in Base Number Systems

  2. Learning Objectives • Add in bases other than 10. • Subtract in bases other than 10. • Multiply in bases other than 10. • Divide in bases other than 10.

  3. Adding in other Bases For the most part, arithmetic operations can be performed in other base number systems just as they are performed in base 10. For example, 9 + 3 = 12 in base 10: when adding 3 to 9, we get past the highest digit in base 10 (which is nine), so we “wrap around” and get one 10, with two 1s left over. In 4five + 3five = 12five, adding in base 10 you get 7, which is past the highest digit in base five (which is four), and so break down 7 to be one 5, with two 1s left over, or 12five.

  4. Adding in other Bases When adding 1-digit numbers in bases other than base ten, add the digits like normal and continue based on the two cases below. Case I: If the result is less than the base, then your answer is complete. Case II: If the result is greater than the base, convert the numeral to the appropriate base. When working in a different base, you can build an addition table that displays the sums when adding one digit numbers. A table for base five is shown here.

  5. EXAMPLE 1 Adding in Base Five Add in base five: 324five + 24five. SOLUTION Step 1 Add the digits in the ones column, 4 and 4. 4five + 4five = 13five (see the table). Write the 3 in the ones place; the digit 1 represents 1 five, so we carry it to the fives column. Step 2 Add 1five + 2five + 2five = 10five (see the table). Write the 0 and carry the 1 to the 25s column as shown. Step 3 Add 1five + 3five = 4five (see the table). Write the 4 as shown. 1 1 0 4 The sum in base five is 324five + 24five = 403five.

  6. EXAMPLE 2 Adding in Base Five Add in base five: 1244five + 333five. SOLUTION The sum is 2132five.

  7. EXAMPLE 3 Adding in Base Two Add in base two: 10111two + 110two. SOLUTION The addition table for base two is The sum is 11101two.

  8. Hexadecimal System The addition table for the hexadecimal system is shown here. Remember that in base 16, the digit A corresponds to 10, B to 11, C to 12, D to 13, E to 14, and F to 15.

  9. EXAMPLE 4 Adding in Base 16 Add in base 16: 135Esixteen + 21Csixteen. SOLUTION The sum is 157Asixteen.

  10. Subtracting In Other Bases Addition tables can help with subtraction; the addition table for base five is below. To perform a subtraction like 12five - 4five, we find 4 in the first column of the table, then move across that row until we find 12. The number at the top of the column is the difference: 12five - 4five = 3five. Numbers that don’t appear in the table will have to be subtracted using a different method.

  11. EXAMPLE 5 Subtracting in Base Five Subtract in base five: 321five - 123five. 1 1 SOLUTION Step 1 Since 3 is larger than 1, it is necessary to borrow a one from the next column; change 2 in the fives column to a 1 and the 1 in the ones column to an 11. This makes the subtraction 11five - 3five = 3five.

  12. EXAMPLE 5 Subtracting in Base Five Subtract in base five: 321five + 123five. 1 1 1 2 SOLUTION Step 2 In the second column 1five - 2five requires borrowing; change 3 in the third column to 2 and take 11five - 2five = 4five. Step 3 Subtract 2five - 1five = 1five. 4 1 The difference is 143five.

  13. Multiplying in Base 5 The multiplication table for base five is shown. For example, 3five x 4five = 22five (3 x 4 = 12 in base ten, which is 22five). To create the table, multiply each pair of numbers in base ten and then convert the answer to base 5.

  14. EXAMPLE 6 Multiplying in Base Five and Base Two (a) Multiply in base five: (b) Multiply in base two:

  15. EXAMPLE 6 Multiplying in Base Five and Base Two (a) First, we multiply each digit in 314five by the last digit in 23five. 2 SOLUTION Step 1 Multiply 4five x 3five = 22five. Write the second digit and carry the first to the fives column.

  16. EXAMPLE 6 Multiplying in Base Five and Base Two SOLUTION Step 2 Multiply 1five x 3five to get 3five and then add the carried 2five to get 10five. Write the 0 and carry the 1 to the 25s column. 1 2 0

  17. EXAMPLE 6 Multiplying in Base Five and Base Two SOLUTION Step 3 Multiply 3five x 3five to get 14five and then add the carried 1five to get 20five. Now, repeat, multiplying each digit in 314five by the first digit in 23five. 1 2 2 0 0 Means add 1+4 first, then write the 0 and carry the 1. Then 1+1=2. 14five +1five 20five

  18. EXAMPLE 6 Multiplying in Base Five and Base Two SOLUTION Step 4 Multiply 4five x 2five to get 13five. Write the second digit in fives column and carry the 1. Step 5 Multiply 1five x 2five to get 2five and add the carried 1 to get 3five. Step 6 Multiply 3five x 2five to get 11five. 1 11 3 Put a zero in the ones column.

  19. EXAMPLE 6 Multiplying in Base Five and Base Two SOLUTION Step 7 Add the partial products in base five. The product is 314five x 23five = 13332five.

  20. EXAMPLE 6 Multiplying in Base Five and Base Two (b) Multiply in base two: 1011two x 11two SOLUTION The multiplication table for base two is

  21. Dividing in Base 5using a Multiplication Table The multiplication table for base five is shown. 1. Find the row in your multiplication table that matches your divisor. (i.e. 11 ÷ 3) 2. Follow along the row in your table to evaluate how many times this numeral divides into the dividend, or the first portion of the dividend. This will be the first digit in the quotient. 3. Write the amount from the table under your dividend and subtract. 4. Bring down the next digit and continue the process, until finished.

  22. EXAMPLE 7 Dividing in Base Five Divide in base five: SOLUTION Step 1 Using the multiplication table for base five, we need to find a product less than or equal to 20five that is divisible by 3five. The number we need is 14five and 3five x 3five = 14five. The first digit in the quotient is 3five.

  23. EXAMPLE 7 Dividing in Base Five 2 SOLUTION Step 2 Then multiply 3five x 3five = 14five and write the product under 20. Subtract 20five - 14five to get 1five, then bring down the next digit. Step 3 Next find a product smaller than or equal to 13five in the table. It is 11five. Since 3five x 2five = 11five, write the 2 in the quotient and write the 11 below the 13. Subtract 13five - 11five, which is 2five, and then bring down the 2. 11 2 2

  24. EXAMPLE 7 Dividing in Base Five 2 4 SOLUTION Step 4 Find a product in the multiplication table divisible by 3five that is less than or equal to 22five. Since 3five x 4five = 22five, write the 4 in the quotient and the 22 below the 22 in the problem. Subtract. 11 2 2 22 The remainder is 0. So 2032five ÷ 3five = 324five. 0

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