Learning to Count

We use base 10 every day — ten symbols, and we carry over after 9. But computers store data in base 2 (binary), display memory in base 16 (hex), and send file attachments in base 64.

What makes a number system? Let's build one from scratch and find out.

In base 10 we have symbols 0–9. After 9 we write 10. In base 7 we only have 0–6. After 6 we write 10. In base 13 we need extra symbols: 0–9 A B C.

Pick any base from 2 to 15 — this will be your number system:

drag to choose

7

your symbols

0 1 2 3 4 5 6

2

Count to 50

Fill in the first 50 numbers in base . Each row has columns — one per symbol. The cell at position n should contain the value n written in your base.

You already know what 0 looks like. What comes right after your last symbol? That is your first two-digit number. Use only the symbols shown on the previous page.

3

Think About It

You've counted in base . Now think about the structure beneath the counting.

Q1 — What is the smallest base possible, and why?

If you only have the symbol 0, can you write a number greater than zero? What is the minimum number of distinct symbols you need to be able to "carry"?

Q2 — In base 10, how many numbers can be represented with exactly 4 digits (counting leading zeros like 0007)?

Each digit position can independently hold any of the 10 symbols. 4 independent choices of 10…

Q3 — In base 10, how many numbers fit in d digits? Write a formula.

Generalise Q2: replace the fixed 4 with d.

Q4 — In base 7, how many numbers fit in d digits? Write a formula.

Same idea as Q3, but now each digit has 7 choices instead of 10.

Q5 — In base b, how many numbers fit in d digits? Write a formula.

Combine Q3 and Q4: replace the fixed base with b.

4

Single-Digit Addition

Fill in the addition table for base . Add two single-digit numbers and write the sum in your base. The result might be a two-digit number when there is a carry.

Example in base 7: 4 + 5 = 9 in decimal, but 12 in base 7 — that's one seven plus two.

Multi-Digit Addition

Now add larger numbers. Work on paper first — carry just like in base 10, but in your base.

5

Subtraction

Subtraction in base works like base 10 — but when you need to borrow, you borrow a group of , not 10.

Example in base 7: 20 − 4. You can't subtract 4 from 0, so you borrow: the left digit decreases by 1, the right digit becomes 0 + 7 − 4 = 3. Answer: 13.

Single-Digit Subtraction

Subtract smaller from larger. Write the result in your base.

Multi-Digit Subtraction

Use borrowing. Write the result in your base.

6

Multiplication

Fill in the multiplication table for base . Multiply two single-digit numbers and write the product in your base.

Multi-Digit Multiplication

Use your multiplication and addition tables. Work column by column, just like long multiplication in base 10.

✓

You Invented It!

You built a complete working number system in base from first principles. Here is what you derived:

Symbol set Positional place-value rule Carry in addition Borrow in subtraction Multiplication table b^d capacity formula

The rules you found — carry, borrow, b^d combinations — are exactly how binary, octal, and hexadecimal work inside every computer. The only thing that changes is the base.

Try a different base and see how the tables reshape.