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**EinsteinJr10****Member**- Registered: 2009-11-08
- Posts: 1

Knowing that ϕ²=ϕ−1, find ϕ-²

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

ϕ[sup]2[/sup] = ϕ - 1

ϕ = 1 - ϕ[sup]-1[/sup]

ϕ[sup]-1[/sup] = 1 - ϕ

ϕ[sup]-2[/sup] = (1-ϕ)[sup]2[/sup]

ϕ[sup]-2[/sup] = 1 - 2ϕ + ϕ[sup]2[/sup]

ϕ[sup]-2[/sup] = 1 - 2ϕ + (ϕ - 1)

ϕ[sup]-2[/sup] = -ϕ

This may seem counterintuitive, but it's worth noting that ϕ must be a complex number.

Why did the vector cross the road?

It wanted to be normal.

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**scientia****Member**- Registered: 2009-11-13
- Posts: 224

If *ϕ* is the golden ratio, *ϕ*² should be *ϕ*+1, not *ϕ*−1.

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**jonnyj99****Member**- Registered: 2010-08-16
- Posts: 4

In mathematics and the arts, both quantities are in the golden ratio if the ratio between the total amounts to the largest number is equal to (=) the ratio of highest to lowest amount of Euclid one.Although not use the term, we call Build this golden ratio.First an isosceles triangle whose base angles are twice the angle at the top.

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