Numbers that lie — From Blank File to Backtest

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Worked examples

Read the claim, then run it and check the machine agrees. One at a time — nothing here is taken on trust.

Example 1 of 4 The lie, in full

example.py

print(0.1 + 0.2)                 # not 0.3...
print((0.1 + 0.2) == 0.3)        # ...so this is a shock
print(format(0.1, ".17f"))       # the hidden 17th digit

Line 1 prints 0.30000000000000004 — the rounding error made visible. Line 2 prints False: the stored sum differs from the stored 0.3 in the last place, and == reports that honestly. Line 3 prints 0.10000000000000001 — proof that even a lone 0.1 was never exact; Python just hides the tail when it prints normally.

Example 2 of 4 Ten dimes, and the fix

example.py

import math

total = 0.0
for _ in range(10):
    total = total + 0.1        # add a tenth, ten times

print(total)                   # should be 1.0 ... right?
print(total == 1.0)            # the trap
print(math.isclose(total, 1.0))  # the honest question

The loop adds 0.1 ten times, so total should be 1.0. It prints 0.9999999999999999 — each add carried a speck of rounding and they accumulated. total == 1.0 is therefore False. But math.isclose(total, 1.0) is True: it asks "close enough, allowing for float rounding?" and that is the question you actually meant.

Example 3 of 4 Decimal keeps the cents

example.py

from decimal import Decimal

print(Decimal("0.1") + Decimal("0.2"))              # exact
print(Decimal("0.1") + Decimal("0.2") == Decimal("0.3"))
print(Decimal("19.99") * 3)                          # three items

Decimal counts in tens like a human, so Decimal("0.1") + Decimal("0.2") is exactly Decimal('0.3'), and comparing it with Decimal("0.3") is True — the thing that failed for floats. The last line, three items at 19.99, gives Decimal('59.97'), exact to the cent. (Pass Decimal strings, not floats, or you'd hand it the rounded value to begin with.)

Example 4 of 4 Big whole numbers stay exact

example.py

print(2 ** 100)                       # integers: no size limit
print(2.0 ** 53 == 2.0 ** 53 + 1)     # floats: precision runs out

2 ** 100 prints 1267650600228229401496703205376 — every one of its 31 digits correct, because Python integers are exact and unbounded. The float line prints True: above 2 ** 53 the representable numbers have gaps, so 2.0 ** 53 + 1 rounds back to 2.0 ** 53 and the two compare equal. Same arithmetic, two very different guarantees.

Check the concept

One question at a time. Unsure? Revisit the lecture, then answer.

Question 1 of 4

What does `0.1 + 0.2 == 0.3` evaluate to, and why?

True — they're obviously the same number False — the stored values differ by a tiny rounding error It raises an error; you can't compare floats True, because Python rounds the result to 0.3 first

The challenge

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