2.2: Addition in Alphabitia (and Base-5)

nataliehobson; Iain Pardoe

2.2: Addition in Alphabitia (and Base-5)

Addition in Alphabitia (and Base-5)

Introduction

Recall, in Alphabitia, numbers are written in base-5 using the symbols:

0, A, B, C, D

Here’s the correspondence with base-10 numbers we’re more familiar with:

Alphabitia Symbol	Base-10 Value
0	0
A	1
B	2
C	3
D	4

When Alphabitians need to count beyond D, they create a new place value (just like in base-10, after 9 we start a new place with 10).

For example:

D (4 in base-10)
A0 (next after D, which equals 5 in base-10)
AA (which equals 6 in base-10)

So Alphabitia works on the same principles as base-10, but instead of grouping by tens, they group by fives.

Review: Place Value in Alphabitia

In base-5, each place represents powers of 5:

Place	…	5²	5¹	5⁰
Value	…	25	5	1

Examples: What is the Alphabitia number C0A in base-10?

Let’s start by interpreting each digit in the number COA.

C = 3 (in the 25s place → 3 × 25 = 75)
0 = 0 (in the 5s place → 0 × 5 = 0)
A = 1 (in the 1s place → 1 × 1 = 1)

So C0A = 76 in base-10.

Review: Blocks to Represent Numbers

We can use Alphabitia-style blocks to visualize:

🟪 = a “flat” of 25 (5×5)
| = a “long” of 5
- = a “unit” of 1

Examples: How to represent the number BD using blocks?

We interpret each digit in the number BD.

B = 2 in the 5s place → ||
D = 4 in the 1s place → ****
So BD = 2×5 + 4×1 = 14 in base-10.

And we can represent BD using blocks as ||****

Rules for Addition in Alphabitia

Addition works the same way as in base-10:

Add the digits in each place.
If the sum is D or less , write it down.
If the sum is larger than D or greater, “regroup” (just like carrying in base-10).

Important regrouping facts:

A + D = * + **** → which means we have ***** = | which is ONE long. So A + D = A0.
B + C = ** + *** = ***** → replaces with one long | = A0.
D + D = **** + **** → regroup as | ***written as AC.

Examples

See the following video for the teacher working through more explanations and examples:

More Examples:

Remember the important thing here is to RE-GROUP!

Example 1: C + C

C = 3, C = 3 → total = 6
6 is bigger than 4, so regroup: 6 = 1 group of 5 + 1 left over
In Alphabitia: C + C = 11

Example 2: BD + AC

Let’s add:

Step 1: Add ones place → D + C

D = 4, C = 3 → total = 7
7 = 1 group of 5 + 2 left over
This is A in the long place, and B in the ones place.

Step 2: Add fives place → B + A + 1 (carry)

B = 2, A = 1 → total = 3 longWrite D

Final Answer: DB

Check in base-10:

BD = 14
AC = 8
14 + 8 = 22
DB (base-5) = 22 ✔️

Example 3: 2-Digit Addition with Bigger Carry

Step 1: Ones → D + D = 8

8 = 1 group of 5 + 3 left over
Write C, carry 1

Step 2: Fives → D + D + 1 = 9

9 = 1 group of 5 + 4 left over
Write D, carry 1

Final Answer: 1DC