Adding and Subtracting in Other Bases

Learning Outcomes

  • Add two numbers from a base other than 10
  • Subtract two numbers from a base other than 10

Adding Basic Time Units

Think about the number of seconds in a minute and minutes in an hour.  What base does this follow?  If you said sixty, you are correct! Further, if you add 2 hours 20 minutes 35 seconds and 1 hour 45 minutes  50 seconds, is the answer 3 hours 65 minutes 85 seconds?  Technically, this is true, but we wouldn’t keep this format.  Once we reach 60 or higher in an unit, we convert 60 it to one in the next higher unit.  You would follow these steps, 65 minutes is equal to 1 hour 5 minutes and 85 seconds is equal to 1 minute 15 seconds.  Thus, 3 hours 65 minutes 85 seconds is roughly 3 hours + 1 hour 5 minutes + 1 minute 15 seconds and simplifies to 4 hours 6 minutes 15 seconds.  Adding non-decimal (not base 10) numbers uses similar steps to adding different time units.

Understanding the Process of Adding in Decimal Numbers

Before we try adding in other bases let’s review the steps to adding two base 10 numbers.  Add these two numbers without a calculator:

536
+ 398

This problem requires carrying to get the correct answer.  Majority of the population can show the steps necessary to get the answer to the above addition problem, but fewer can explain why we carry?  This video answers the question of why carrying works and will prepare you for adding in a base other than 10.

Adding in Other Bases

When you think back to when you first learned addition, it is very likely you learned the addition table. Once you knew the addition table, you moved on to addition of numbers with more than one digit. The same process holds for addition in other bases. We begin with an addition table, and then move on to adding numbers with two or more digits.

Here’s the beginning of the base 6 addition table:

+ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 ?
2 2 3 4 5 ? ?
3 3 4 5 ? ? ?
4 4 5 ? ? ? ?
5 5 ? ? ? ? ?

Many of the cells are not filled out. The ones filled in are values that never get past 5, which is the largest legal symbol in base 6, so they are acceptable symbols. But what do we do with 5 + 3 in base 6? We can’t represent the answer as “8” since “8” is not a symbol available to us. Let’s look at the list of first 36 numbers we have for base 6.

0 1 2 3 4 5
10 11 12 13 14 15
20 21 22 23 24 25
30 31 32 33 34 35
40 41 42 43 44 45
50 51 52 53 54 55

So, what is 5 + 1 equal to in base 6? Well, start at the 5, and jump ahead one step. You land on 10.

A table with numbers 0 to 5, 10 to 15, 20 to 25, 30 to 35, 40 to 45, 50 to 55. An arrow points from 5 to 10
A table with numbers 0 to 5, 10 to 15, 20 to 25, 30 to 35, 40 to 45, 50 to 55. A callout pointing to 10 reads, The 10 is one step past the 5.
This means that, in base 6, 5 + 1 = 10.

So, what is 5 + 2 in base 6? Well, 5 + 2 = 5 + 1 + 1, so 10 + 1…jump one more space and you land on 11. So, 5 + 2 = 11 in base 6.

A table with numbers 0 to 5, 10 to 15, 20 to 25, 30 to 35, 40 to 45, 50 to 55. Arrows point from 5 to 10 and from 10 to 11.
And so it goes. Using that process, stepping one more along the list, we can fill in the remainder of the base 6 addition table.
+ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 10
2 2 3 4 5 10 11
3 3 4 5 10 11 12
4 4 5 10 11 12 13
5 5 10 11 12 13 14

With this table, and with our understanding of “carrying the one,” we can then use the addition table to do addition in base 6 for numbers with two or more digits, using the same processes you learned for addition when you did it by hand.

Example

Adding in Base 6

Calculate 2516 + 1336.

Try It

Calculate 4536 + 3456.

To summarize the creation of the addition tables for a given base, do the following.

Step 1: Set up the table.

Step 2: Fill in all the additions that use the “legal” symbols for the base. The diagonal that goes from upper left to lower right that is immediately next to the filled boxes all get the value 10, regardless of base.

Step 3: Enter the values that are in the “teens.” This can all be done on one table without creating multiple copies of previously done work.

Examples

Creating an Addition Table for a Base Lower Than 10

  1. Create the addition table for base 7.
  2. Create the addition table for base 2.

Try It

Create the addition table for base 4.

Example

Adding in Base 7

Calculate 5367 + 4337.

Try It

Calculate 4617 + 1427.

As seen previously, when performing addition in another base, set up the problem exactly as you would for addition in base 10. At each step, check the addition table for the base. As in base 10 addition, move right to left, adding down the columns using the rules in the addition table. When necessary and just as in base 10, be sure to carry the 1.

The following video shows how to add in base 5 and 8.

Example

Adding in Base 2 – Binary System

We return to base 2, the base used by computers.  Calculate 10012 + 110112.

Try It

Calculate 11000112 + 1011112.

Example

Creating an Addition Table for a Base Higher Than 10

Create the addition table for base 12.

Example

Adding in Base 12

Calculate 3A712 + 9BA12.

Try It

Calculate 4B312 + B0612.

Subtracting in Other Bases

Sometimes carrying is necessary when adding numbers. Likewise, sometimes borrowing is necessary when subtracting numbers.  As you view these two videos, pay close attention to the steps and understanding what it means to borrow.

Subtraction in bases other than base 10 follow the same processes as base 10 subtraction, but, as with addition, using the addition table for the base.

Example

Subtracting in Base 6

Calculate 526 − 346.

Try It

Calculate 1156 − 436.

You will see two different approaches to representing the value borrowed:
1. Writing the borrowed amount by adding the digit 1 (like above example) keeping all values in the problem in the given base.
2. Writing the borrowed amount in base 10 since it’s “scratch paper” as in the videos below.