In mathematics, Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} , sometimes called the golden field, is a number system consisting of the set of all numbers a + b 5 {\displaystyle a+b{\sqrt {5}}} , where a {\displaystyle a} and b {\displaystyle b} are both rational numbers and 5 {\displaystyle {\sqrt {5}}} is the square root of 5, along with the basic arithmetical operations (addition, subtraction, multiplication, and division). This arithmetic shares certain structural properties with the arithmetic of Q {\displaystyle \mathbb {Q} } , the field of rational numbers, making Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} also a type of field. More specifically, it is a real quadratic field, the extension field of Q {\displaystyle \mathbb {Q} } generated by combining rational numbers and 5 {\displaystyle {\sqrt {5}}} using arithmetical operations. The name comes from the golden ratio φ {\displaystyle \varphi } , a positive number satisfying the equation φ 2 = φ + 1 {\displaystyle \textstyle \varphi ^{2}=\varphi +1} , which is the fundamental unit of Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} . Calculations in the golden field can be used to study the Fibonacci numbers and other topics related to the golden ratio, notably the geometry of the regular pentagon and higher-dimensional shapes with fivefold symmetry.
Basic arithmetic Elements of the golden field are those numbers which can be written in the form a + b 5 {\displaystyle a+b{\sqrt {5}}} where a {\displaystyle a} and b {\displaystyle b} are uniquely determined rational numbers, or in the form ( a + b 5 ) / c {\displaystyle {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}{\big /}c} where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are integers, which can be uniquely reduced to lowest terms, and where 5 = 2.236 … {\displaystyle {\sqrt {5}}=2.236\ldots } is the square root of 5. It is sometimes more convenient instead to use the form a + b φ {\displaystyle a+b\varphi } where a {\displaystyle a} and b {\displaystyle b} are rational or the form ( a + b φ ) / c {\displaystyle (a+b\varphi )/c} where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are integers, and where φ = 1 2 ( 1 + 5 ) =
{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}={}\!}
1.618 … {\displaystyle 1.618\ldots } is the golden ratio. Converting between these alternative forms is straight-forward: a + b 5 = ( a − b ) + ( 2 b ) φ {\displaystyle a+b{\sqrt {5}}=(a-b)+(2b)\varphi } or, in the other direction, a + b φ = ( a + 1 2 b ) + ( 1 2 b ) 5 {\displaystyle a+b\varphi ={\bigl (}a+{\tfrac {1}{2}}b{\bigr )}+{\bigl (}{\tfrac {1}{2}}b{\bigr )}{\sqrt {5}}} . To add or subtract two numbers, simply add or subtract the components separately:
( a 1 + b 1 5 ) + ( a 2 + b 2 5 ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) 5 , ( a 1 + b 1 φ ) + ( a 2 + b 2 φ ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) φ . {\displaystyle {\begin{aligned}{\bigl (}a_{1}+b_{1}{\sqrt {5}}~\!{\bigr )}+{\bigl (}a_{2}+b_{2}{\sqrt {5}}~\!{\bigr )}&=(a_{1}+a_{2})+(b_{1}+b_{2}){\sqrt {5}},\\[3mu](a_{1}+b_{1}\varphi )+(a_{2}+b_{2}\varphi )&=(a_{1}+a_{2})+(b_{1}+b_{2})\varphi .\end{aligned}}}
To multiply two numbers, distribute:
( a 1 + b 1 5 ) ( a 2 + b 2 5 ) = ( a 1 a 2 + 5 b 1 b 2 ) + ( a 1 b 2 + b 1 a 2 ) 5 , ( a 1 + b 1 φ ) ( a 2 + b 2 φ ) = ( a 1 a 2 + b 1 b 2 ) + ( a 1 b 2 + b 1 a 2 + b 1 b 2 ) φ . {\displaystyle {\begin{aligned}{\bigl (}a_{1}+b_{1}{\sqrt {5}}~\!{\bigr )}{\bigl (}a_{2}+b_{2}{\sqrt {5}}~\!{\bigr )}&=(a_{1}a_{2}+5b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2}){\sqrt {5}},\\[3mu](a_{1}+b_{1}\varphi )(a_{2}+b_{2}\varphi )&=(a_{1}a_{2}+b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2}+b_{1}b_{2})\varphi .\end{aligned}}}
To find the reciprocal of a number α {\displaystyle \alpha } , rationalize the denominator: 1 / α =
{\displaystyle 1/\alpha ={}}
α ¯ / α α ¯ =
{\displaystyle {\overline {\alpha }}/\alpha {\overline {\alpha }}={}}
α ¯ / N ( α ) {\displaystyle {\overline {\alpha }}/\mathrm {N} (\alpha )} , where α ¯ {\displaystyle {\overline {\alpha }}} is the algebraic conjugate and N ( α ) {\displaystyle \mathrm {N} (\alpha )} is the field norm, as defined below. Explicitly:
1 a + b 5 = 1 a + b 5 ⋅ a − b 5 a − b 5 = a a 2 − 5 b 2 − b a 2 − 5 b 2 5 , 1 a + b φ = 1 a + b φ ⋅ a + b − b φ a + b − b φ = a + b a 2 + a b − b 2 − b a 2 + a b − b 2 φ . {\displaystyle {\begin{aligned}{\frac {1}{a+b{\sqrt {5}}}}&={\frac {1}{a+b{\sqrt {5}}}}\cdot {\frac {a-b{\sqrt {5}}}{a-b{\sqrt {5}}}}={\frac {a}{a^{2}-5b^{2}}}-{\frac {b}{a^{2}-5b^{2}}}{\sqrt {5}},\\[3mu]{\frac {1}{a+b\varphi }}&={\frac {1}{a+b\varphi }}\cdot {\frac {a+b-b\varphi }{a+b-b\varphi }}={\frac {a+b}{a^{2}+ab-b^{2}}}-{\frac {b}{a^{2}+ab-b^{2}}}\varphi .\end{aligned}}}
To divide two numbers, multiply the first by second's reciprocal:
a 1 + b 1 5 a 2 + b 2 5 = a 1 a 2 − 5 b 1 b 2 a 2 2 − 5 b 2 2 + − a 1 b 2 + b 1 a 2 a 2 2 − 5 b 2 2 5 , a 1 + b 1 φ a 2 + b 2 φ = a 1 a 2 + a 1 b 2 − b 1 b 2 a 2 2 + a 2 b 2 − b 2 2 + − a 1 b 2 + b 1 a 2 a 2 2 + a 2 b 2 − b 2 2 φ . {\displaystyle {\begin{aligned}{\frac {a_{1}+b_{1}{\sqrt {5}}}{a_{2}+b_{2}{\sqrt {5}}}}&={\frac {a_{1}a_{2}-5b_{1}b_{2}}{a_{2}^{2}-5b_{2}^{2}}}+{\frac {-a_{1}b_{2}+b_{1}a_{2}}{a_{2}^{2}-5b_{2}^{2}}}{\sqrt {5}},\\[6mu]{\frac {a_{1}+b_{1}\varphi }{a_{2}+b_{2}\varphi }}&={\frac {a_{1}a_{2}+a_{1}b_{2}-b_{1}b_{2}}{a_{2}^{2}+a_{2}b_{2}-b_{2}^{2}}}+{\frac {-a_{1}b_{2}+b_{1}a_{2}}{a_{2}^{2}+a_{2}b_{2}-b_{2}^{2}}}\varphi .\end{aligned}}}
As in any field, addition and multiplication of numbers in Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} is associative and commutative; 0 {\displaystyle 0} is the additive identity and 1 {\displaystyle 1} is the multiplicative identity; every number α {\displaystyle \alpha } has an additive inverse − α {\displaystyle -\alpha } and a multiplicative inverse 1 / α {\displaystyle 1/\alpha } ; and multiplication distributes over addition. Arithmetic between numbers in Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} is consistent with their arithmetic as real numbers; that is, Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} is a subfield of R {\displaystyle \mathbb {R} } .
Conjugation and norm The numbers 5 {\displaystyle {\sqrt {5}}} and − 5 {\displaystyle -{\sqrt {5}}} each solve the equation x 2 = 5 {\displaystyle \textstyle x^{2}=5} . Each number α = a + b 5 {\displaystyle \alpha =a+b{\sqrt {5}}} in Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} has an algebraic conjugate α ¯ {\displaystyle {\overline {\alpha }}} found by swapping these two square roots of 5, i.e., by changing the sign of b {\displaystyle b} . The conjugate of φ {\displaystyle \varphi } is φ ¯ = 1 2 ( 1 − 5 ) =
{\displaystyle {\overline {\varphi }}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}={}}
− φ − 1 =
{\displaystyle \textstyle -\varphi ^{-1}={}}
1 − φ {\displaystyle 1-\varphi } . A rational number is its own conjugate, a = a ¯ {\displaystyle a={\overline {a}}} . In general, the conjugate is:
a + b 5 ¯ = a − b 5 , a + b φ ¯ = a + b φ ¯ = ( a + b ) − b φ . {\displaystyle {\begin{aligned}{\overline {a+b{\sqrt {5}}}}&=a-b{\sqrt {5}},\\[3mu]{\overline {a+b\varphi }}&=a+b{\overline {\varphi }}=(a+b)-b\varphi .\end{aligned}}}
Conjugation in Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} is an involution, ( α ¯ ) ¯ = α {\displaystyle {\overline {({\overline {\alpha }})}}=\alpha } , and it preserves the structure of arithmetic: α 1 + α 2 ¯ = α ¯ 1 + α ¯ 2 {\displaystyle {\overline {\alpha _{1}+\alpha _{2}}}={\overline {\alpha }}_{1}+{\overline {\alpha }}_{2}} ; α 1 α 2 ¯ = α ¯ 1 α ¯ 2 {\displaystyle {\overline {\alpha _{1}\alpha _{2}}}={\overline {\alpha }}_{1}{\overline {\alpha }}_{2}} ; and α 1 / α 2 ¯ = α ¯ 1 / α ¯ 2 {\displaystyle {\overline {\alpha _{1}/\alpha _{2}}}={\overline {\alpha }}_{1}/\,{\overline {\alpha }}_{2}} . Conjugation is the only ring homomorphism (function preserving the structure of addition and multiplication) from Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} to itself, other than the identity function. The field trace is the sum of a number and its conjug