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# Why is a==b False?

Please see the following code. Why is a==b False? Thanks!

Input:

a=n(1/2*sqrt(5) - 1/2,digits=15)
b=n(1-a^2,digits=15)
print a
print b
a==b


Output:

0.618033988749895
0.618033988749895
False

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## 1 answer

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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
sage: a.radical_expression()
1/2*sqrt(5) - 1/2

more

## Comments

3

Note that even without defining the corresponding number field or working in QQbar, SageMath correctly finds that a == b if you are not working with approximations but with exact symbolic expressions¹:

sage: a = 1/2*sqrt(5) - 1/2
sage: b = 1 - a^2
sage: bool(a == b)
True


¹ Note the necessity of calling bool(a==b) since a == b is kept as an expression by default, and one has to explicitly asks whether this equality holds.

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Asked: 2017-12-08 17:45:51 -0500

Seen: 32 times

Last updated: Dec 09 '17