Multiplication of Decimal Numbers

Since decimal numbers are implicitly just polynomials in the powers of 10, e.g.,

$$3819 = 3\cdot 10^3 + 8\cdot 10^2 + 1\cdot 10^1 + 9\cdot 10^0,$$

it follows that multiplying two numbers convolves their digits. The only twist is that, unlike normal polynomial multiplication, we have carries. That is, when a convolution result (output digit) exceeds 10, we subtract 10 from the result and add 1 to the digit in the next higher place.^7.8