Presented by: Tutorial Services The Math Center Number Bases Presented by: Tutorial Services The Math Center

Base 10 to Different Number Bases When converting a number from base 10 to another base follow these steps: Take the number you want to convert to a different base and divide by the new base. When dividing, write the answer with the remainder. Take the quotient, not the remainder, and again divide the new number by the base. Write your answer with the remainder. Next, follow the third step again. Repeat steps 3 and 4 until you get to an answer where you can no longer divide the quotient by the base. For example if 1 is the quotient and 2 is the base, 1 / 2 will not divide into a whole number. You stop here. To write your answer, take the last number you get as a quotient when doing the division and write the remainders next to it starting from the last remainder you got, to the first remainder. Don’t forget to put the base as a subscript at the end of the number.

Example Convert 45 to base 2. 45 / 2 22 1 22 / 2 11 0 11 / 2 5 1 Quotient Remainder 45 / 2 22 1 22 / 2 11 0 11 / 2 5 1 5 / 2 2 1 2 / 2 1 0 Stop --> 1 / 2 Take the last quotient you got before you stopped, and that would be the first digit of the new number in base 2. Next, write the remainders starting from the last remainder you got, to the first remainder. Answer: 45 = 1011012

Exercise Convert 94 to base 6. Solution: 94 / 6 15 4 15 / 6 2 3 Quotient Remainder 94 / 6 15 4 15 / 6 2 3 Stop -> 2/ 6 Answer: 94 = 2346

Numbers in Different Bases to Base 10 When converting numbers in different bases to base 10 follow these steps: First count the amount of digits in the number you will process. (i.e. 25 has 2 digits) Then, take the first number going from left to right and multiply it by the base it is on raised to one number less than the total numbers of digits in the number. (i.e. 202 has 2 digits, therefore the first operation should look like this (2 x 2^1), where 2 is the first digit in the number 202 and 2^1 is the base being raised to its corresponding power.) Keep doing this subsequently with the following numbers. Note that each subsequent digit needs to be multiplied by the base raised to one power less then the previous operation. Keep in mind the last digit will need to be multiplied by 2^0. Finally, add all the results from steps 2 and 3 to get your result in base 10.

Example Convert 1011012 to base 10 The number of digits in the number 1011012 is equal to 6, therefore the first operation should start with a power of 5. (1 x 25) + (0 x 24) + (1 x 23) + (1 x 22) + (0 x 21) + (1 x 20) (1 x 32) + (0 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (1 x 1) 32 + 0 + 8 + 4 + 0 + 1 45

Exercise Convert 2346 to base 10 Solution: (2 x 62) + (3 x 61) + (4 x 60) (2 x 36) + (3 x 6) + (4 x 1) 72 + 18 + 4 94

Addition and Subtraction in Other Bases When adding numbers in different bases follow these steps: First add the numbers as you would normally do it. Ask yourself, how many times does the base go into the sum. If the base does not go into the sum, place the resultant of the sum as part of the answer. If the base goes into the sum, place the remainder as part of the answer and carry over the quotient (the number of times the base goes into the sum). Repeat steps 1 and 2 for the subsequent additions.

Addition and Subtraction in Other Bases (cont.) To subtract numbers in different bases follow these rules: Subtract as you would normally do it. If you cannot subtract because the number is smaller than the one you are subtracting, then you will have to borrow from the following number. Borrow the amount of the base and add to the number that needed the borrowing. The number from which you are borrowing will need to decrease by one. If the number from which you are borrowing happens to be zero, then you will have to borrow from the following number. Repeat steps 2 and 3.

Examples Addition: Subtraction:

Exercise(s) Add Subtract Solution: Solution:

Multiplication In Different Bases Multiplication Procedure Multiply as you would normally do. Figure out how many times the base goes into the resultant of the numbers multiplied. The remainder of the previous step goes on the bottom and how many times the base goes into the resultant carries over. (If the resultant is less then the base number, leave the resultant as part of your answer on the bottom where the remainder would normally go.) Keep doing this for every column. Add like you normally would for adding numbers of the same base. Get your answer

Multiplication Examples

Multiplication Exercise Multiply Solution 2349 X 259

Division in Different Bases Division Procedure: Construct a multiplication table with the corresponding base. Divide as you normally would. Combine all the previous rules for addition, subtraction and multiplication in order to arrive to the expected answer.

Division Exercise(s) Multiplication Table Multiplication Table

Number Bases Links Logic Workshop Student Handout Truth Tables Handout