Q1.
Can you explain each of Brahmagupta’s rules in terms of Bela’s Building of Fun, or in terms of a number line?
Answer
Yes — every rule is just a journey in the lift, or a walk on the number line.
Rules for addition
| Brahmagupta's rule | In the Building of Fun / on the number line |
|---|---|
| 1. Positive + positive is positive (2 + 3 = 5) | Go 2 floors up, then 3 more up — you are 5 floors above the ground. |
| 2. Negative + negative is negative (– 2 + (– 3) = – 5) | Go 2 floors down, then 3 more down — you are 5 floors below the ground. |
| 3. Positive + negative: subtract the digits, keep the sign of the bigger (– 5 + 3 = – 2) | Go 5 floors down, then 3 back up. The 3 ups cancel 3 of the downs and 2 downs are left over, so you finish on Floor – 2. |
| 4. A number + its inverse = 0 (2 + (– 2) = 0) | Press + 2 then – 2 and you are back at the Welcome Hall, Floor 0. |
| 5. A number + 0 = the same number (– 2 + 0 = – 2) | Press the button zero times — you stay exactly where you were. |
Rules for subtraction (remember: Target – Starting = Movement needed)
| Brahmagupta's rule | In the Building of Fun / on the number line |
|---|---|
| 1. Smaller positive from larger positive is positive (3 – 2 = 1) | From Floor 2 to Floor 3 you must go up: press + 1. |
| 2. Larger positive from smaller positive is negative (2 – 3 = – 1) | From Floor 3 to Floor 2 you must come down: press – 1. |
| 3. Subtracting a negative is adding the positive (2 – (– 3) = 2 + 3) | From Floor – 3 to Floor 2 you must first climb 3 floors to reach the ground and 2 more after that — 5 floors up in all. |
| 4. A number minus itself is 0 (– 2 – (– 2) = 0) | You are already on Floor – 2 and want to be on Floor – 2 — no button press at all. |
| 5. Number – 0 = the number; 0 – number = its inverse (0 – (– 2) = 2) | From Floor – 2 to the ground floor you must press + 2, the inverse of – 2. |
Why this matters: Brahmagupta wrote these rules in the Brāhma-sphuṭa-siddhānta in 628 CE — the first time in the world that positive numbers, negative numbers and zero were treated as equally real numbers. The lift model of this chapter is just a picture of the same rules, more than 1300 years later.