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Adding Floating-Point Decimals for Fun and Profit

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Many people know that you shouldn’t do decimal calculations, such as those involving U.S. dollars and cents, with the floating-point numbers in most programming languages. This is because decimal numbers can’t be expressed exactly as such floating-point numbers, so you will encounter rounding errors. Famously, with IEEE double-precision floats, 0.1 + 0.2 is not 0.3, but rather, 0.30000000000000004. On the other hand, I use a Python REPL to add up decimal numbers for receipts all the time. Of course, my stakes are lower; I’m summing things in the $1–$100 range and know to manually round the microscopic errors off before copying the sum somewhere. But actually it’s quite often that those errors don’t appear at all. If we sum every pair of multiples of 0.01 up to 1.00 and look at whether the printed result is too large (blue), too small (red), or correct, we get a cool pattern: Figure 1: How floating-point affects summing two multiples of 0.01, up to 1.00 Where does this pattern come from?

2026-08-30 (2527 words) filed under Math, CS Many people know that you shouldn’t do decimal calculations, such as those involving U.S. dollars and cents, with the floating-point numbers in most programming languages. This is because decimal numbers can’t be expressed exactly as such floating-point numbers, so you will encounter rounding errors. Famously, with IEEE double-precision floats, 0.1 + 0.2 is not 0.3, but rather, 0.30000000000000004. On the other hand, I use a Python REPL to add up decimal numbers for receipts all the time. Of course, my stakes are lower; I’m summing things in…

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