# Revision history [back]

This comes from working with numerical approximations.

In the computer, a and b are represented in binary. They are very close but not equal.

sage: a.sign_mantissa_exponent()
(1, 11133510565745312, -54)
sage: b.sign_mantissa_exponent()
(1, 11133510565745310, -54)


But the difference is very small and their decimal expansions to 15 digits coincide.

Sage lets you work with exact algebraic numbers, either by creating a number field:

sage: K.<a> = NumberField(x^2 + x - 1, embedding=0.6)
sage: a.numerical_approx()
0.618033988749895
sage: 1 - a^2
a


or by using the field of algebraic number (QQbar in Sage):

sage: a = QQbar(1/2*sqrt(5) - 1/2)
sage: a
0.618033988749895?
sage: 1 - a^2 == a
True
sage: a.minpoly()
x^2 + x - 1
sage: a.numerical_approx()
0.618033988749895