Alternative Base Addition

Within these castle walls be forged Mavens of Computer Science ...

— Merlin, The Coder

1 + 1 = 10

Dark apple

Dark apple

Prerequisites

• W1011 Number Systems

Addition (Decimal System)

Addition is a basic (and critical) operation. Important properties of addition include:

• Addition is commutative, meaning that the order of the operands does not matter
  • $a+b=b+a$
• Addition is associative, meaning that when we're adding more than two operands, the order that we perform the addition does not matter
  • $(a+b)+c=a+(b+c)$
• The identity element for addition, also termed the additive identity, is zero
  • $a+0=a$
• The operator for addition is the plus ($+$) sign
• The operands for addition are called addends

Let's review how we perform addition in the number system with which we are most familiar, the decimal system.

There are a few simple rules when adding non-negative, whole numbers:

1. Align the addends vertically, flush right. (This step ensures that the position multiplier is the same for both addends in each column.)
2. Starting from the right-most column, add the two digits. If the sum exceeds the quantity that can be represented with a single digit, carry a one to the column to the left.
3. Repeat the process with the column to the left until reaching the final column.

Let's look at a few examples:

To add $3+5$ we: align the digits vertically add the right-most (and only column), yielding 8. Because this is less than the quantity that can be represented by a single digit, we have no carry. done

carry 0 addend 3 addend 5 sum 8

References

• Addition (Wikipedia)