Book IX · Proposition 23

IX.23

If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.Heath, 1908

Every step, checked

• all the numbers are oddgiven
• they are odd in multitudegiven
• the sum is oddby IX.22

What it needs, and what needs it

Needs: IX.22

Used by: IX.25 IX.29

Rests on: Def.VII.6

Depth: 2 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "IX.23",
    THEOREM,
    sample=lambda rng: _odd_numbers(rng, even_count=False),
)
def prop_IX_23(given: list) -> Out:
    hypothesis("all the numbers are odd", all(not measures(2, n) for n in given) and given)
    hypothesis("they are odd in multitude", not measures(2, len(given)))
    because(prop_IX_22, given + [given[0]]) if len(given) % 2 else None

    claim("the sum is odd", "IX.22", not measures(2, sum(given)))
    return Out(total=sum(given))