Reciprocals of some mathematical expressions

Introduction

This page is about simplifying the reciprocals of some mathematical expressions by moving square roots from the denominator to the numerator.

Reciprocals involving the golden ratio

The golden ratio (phi): φ = (1 + √5) / 2 ≈ 1.618

1 /	(`aφ` + `b`)	=	(`a`(`φ` − 1) − `b`)	/	(`a`² − `ab` − `b`²)
1 /	(`aφ` + 1)	=	(`a`(`φ` − 1) − 1)	/	(`a`² − `a` − 1)
1 /	(`φ` + `a`)	=	(`a` + 1 − `φ`)	/	(`a`² + `a` − 1)
1 /	`φ`	=	`φ` − 1
1 /	(`φ` + 1)	=	2 − `φ`

Example: 1 / (5φ + 2) = (5φ − 7) / 11

Other reciprocals

1 /	√`a`	=	√`a`	/	`a`
1 /	(`a` + √`b`)	=	(`a` − √`b`)	/	(`a`² − `b`)
1 /	(`a` − √`b`)	=	(`a` + √`b`)	/	(`a`² − `b`)
1 /	(√`a` + √`b`)	=	(√`a` − √`b`)	/	(`a` − `b`)
1 /	(√`a` − √`b`)	=	(√`a` + √`b`)	/	(`a` − `b`)

Example: 1 / (3 + √7) = (3 − √7) / 2