Algebra of Bhaskara II

Example.

From a positive, subtract a positive; from a negative, subtract a negative; and reverse the signs. Tell me quickly the remainder.
Layout: 3, 2. The difference is 1.
3̇, 2̇. The difference is 1̇.
3̇, 2. The difference is 5̇.
3, 2̇. The difference is 5.

Thus ends the addition and subtraction of positive and negative.

Commentary: The author presents four examples using the first half of the Upajātikā verse.
"From three subtract two" implies: first, from positive 3 subtract positive 2; second, from negative 3 subtract negative 2; third, if reversed, from positive 3 subtract negative 2; fourth, from negative 3 subtract positive 2. These are the four examples.

In the first, the layout is 3 and 2. The subtrahend 2 becomes negative, and the difference is 1.
In the second, the layout is 3̇ and 2̇. The subtrahend 2 (negative) becomes positive, and the difference is 1̇.
In the third, the layout is 3̇ and 2. The subtrahend becomes negative, and the sum is 5̇.
In the fourth, the layout is 3 and 2̇. The subtrahend (negative) becomes positive, and the sum is 5. Thus everywhere.

Vimalā (Commentary): Tell me quickly: from a positive 3, subtract positive 2; from negative 3, subtract negative 2; from positive 3, subtract negative 2; and from negative 3, subtract positive 2. What will be the remainder?

Example —

Difference of 3 and 2 = (3) — (2) = 1,
Difference of 3̇ and 2̇ = (— 3) — (— 2) = — 1,
Difference of 3̇ and 2 = (— 3) — (2) = — 5,
Difference of 3 and 2̇ = (3) — (— 2) = 5,

New types of examples in subtraction:

(1) Subtract y from k + n,
given y = 4, k = 6, n = 8.
Therefore, k + n — y = 6 + 8 — 4 = 14 — 4 = 10.
(2) Subtract 2y² + 3y·k — 5k² from — 3y² + 2y·k — 4k².
Difference = — 3y² + 2y·k — 4k² — 2y² — 3y·k + 5k² =
(— 3y² — 2y²) + (2y·k — 3y·k) + (— 4k² + 5k²) =
— 5y² — y·k + k².
(3) Subtract y³ — 3y·k + 5n³ — p² from 3n³ + 2p² — 7y³.