I need to multiply these two large numbers exactly:
123456789123456789 × 987654321987654321

Let me break this down using the standard multiplication algorithm.

First, let me notice that:
- 123456789123456789 = 123456789 × 10^9 + 123456789
- 987654321987654321 = 987654321 × 10^9 + 987654321

So I can write:
123456789123456789 = 123456789 × (10^9 + 1)
987654321987654321 = 987654321 × (10^9 + 1)

Therefore:
123456789123456789 × 987654321987654321 = 123456789 × 987654321 × (10^9 + 1)^2

First, let me calculate 123456789 × 987654321:
123456789 × 987654321 = 121932631112635269

Now I need to calculate (10^9 + 1)^2:
(10^9 + 1)^2 = 10^18 + 2 × 10^9 + 1

So:
121932631112635269 × (10^18 + 2 × 10^9 + 1)
= 121932631112635269 × 10^18 + 121932631112635269 × 2 × 10^9 + 121932631112635269
= 121932631112635269000000000000000000 + 243865262225270538000000000 + 121932631112635269
= 121932631112635269243865262225270538121932631112635269

Therefore:
123456789123456789 × 987654321987654321 = 121932631112635269243865262225270538121932631112635269
