### The Maxima on-line user's manual

Algebra Calculator

#### Phi

-- Constant: %phi `%phi` represents the so-called golden mean, (1 + sqrt(5))/2. The numeric value of `%phi` is the double-precision floating-point value 1.618033988749895d0.

`fibtophi` expresses Fibonacci numbers `fib(n)` in terms of `%phi`.

By default, Maxima does not know the algebraic properties of

`     `%phi`.  After evaluating `tellrat(%phi^2 - %phi - 1)` and`
`     `algebraic: true`, `ratsimp` can simplify some expressions`
`     containing `%phi`.`

Examples:

`fibtophi` expresses Fibonacci numbers `fib(n)` in terms of `%phi`.

```          (%i1) fibtophi (fib (n));
n             n
%phi  - (1 - %phi)
(%o1)                  -------------------
2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) fibtophi (%);
n + 1             n + 1       n             n
%phi      - (1 - %phi)        %phi  - (1 - %phi)
(%o3) - --------------------------- + -------------------
2 %phi - 1                2 %phi - 1
n - 1             n - 1
%phi      - (1 - %phi)
+ ---------------------------
2 %phi - 1
(%i4) ratsimp (%);
(%o4)                           0```

By default, Maxima does not know the algebraic properties of

`     `%phi`.  After evaluating `tellrat (%phi^2 - %phi - 1)` and`
`     `algebraic: true`, `ratsimp` can simplify some expressions`
`     containing `%phi`.`

```          (%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
2                      2
(%o1)        %phi  A - %phi A - A + %phi  - %phi - 1
(%i2) ratsimp (e);
2                     2
(%o2)        (%phi  - %phi - 1) A + %phi  - %phi - 1
(%i3) tellrat (%phi^2 - %phi - 1);
2
(%o3)                  [%phi  - %phi - 1]
(%i4) algebraic : true;
(%o4)                         true
(%i5) ratsimp (e);
(%o5)                           0```

There are also some inexact matches for `phi`. Try `?? phi` to see them.

```(%o1)                                true
(%i2) ```

### Related Examples

##### phi-plot2d-sqrt

L: 1;

phi(z):= sqrt(105/(2*...

phi(-1/2);

Calculate

##### phi-plot2d

phi: x^2+y^2-100;

plot2d(phi,[x,-20,20]...

Calculate

##### phi-tan

eq1:V=h*pi/3*(R^2+R*r...

eq2:R=r+h*tan(phi);

eq3:h=(R-r)/tan(phi);

Calculate

##### phi

fourier (f, x, phi);

Calculate

##### phi-sin-sqrt-trigreduce

2*sin(phi)*sqrt(1-sin...

trigreduce(%);

Calculate

phi: %pi / 6;

l1: 0.3;

l2: 0.8;

Calculate

##### phi-sin-solve

S : (sin(phi)-sin(phi...

solve([s=S],[phi]);

Calculate

##### phi-sinh-taylor

c: (1/xi)*sinh(xi*phi...

taylor(c, phi, 0, 4);

Calculate

m:12.5E-3;

M:240E-3;

k:2.25E+3;

Calculate

##### phi-sin-solve

S : (sin(phi)-sin(phi...

solve([s=S],[phi]);

Calculate