Musings on the HP-70

[Thomas Klemm]

---

Fibonacci Sequence

Initialisation

DSP 0
CLR
STO M
1

Loop

M+
x<>y


Result

0.
1.
1.
2.
3.
5.
8.
13.
21.
34.


Explanation

\(
\begin{aligned}
x_{0} &= 0 \\
x_{1} &= 1 \\
\\
x_{n+1} &= x_{n} + x_{n-1} \\
\end{aligned}
\)

Python Program

Code:

a, b = 0, 1
for k in range(10):
    print(a)
    a, b = b, a + b

References

• Fibonacci sequence
  

---

Viète's formula for \(\pi\)

Initialisation

DSP 9
0.5
STO K
CLR
STO M
2
ENTER
ENTER
ENTER

Loop

x<>y
M+
K
y^x
STO M
÷
×


Result

2.000000000
2.828427125
3.061467459
3.121445152
3.136548491
3.140331157
3.141277251
3.141513801
3.141572940
3.141587725
3.141591422
3.141592346
3.141592577
3.141592634
3.141592649
3.141592652
3.141592653
3.141592654
3.141592654


Explanation

\(
\pi = 2 \cdot \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{2 + \sqrt{2}}} \cdot \frac{2}{\sqrt{2 + \sqrt{2 + \sqrt{2}}}} \cdots
\)

Python Program

Code:

from math import sqrt

p, q = 2, 0
for k in range(20):
    print(f"{p:>.9f}")
    q = sqrt(2 + q)
    p *= 2 / q

References

• Viète's formula
  

---

• Can you guess the result?
• Can you come up with other interesting recipes?
  

[Thomas Klemm]

Here's another one:

---

Euler's number

Initialisation

DSP 9
-1
STO K
13
STO M
1
ENTER
ENTER
ENTER

Loop

K
M+
÷
+

Result

1.000000000
1.083333333
1.098484848
1.109848485
1.123316498
1.140414562
1.162916366
1.193819394
1.238763879
1.309690970
1.436563657
1.718281828
2.718281828


Explanation

\(
\begin{aligned}
e
&= 1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots \\
&= 1 + \frac{1}{1}\left(1 + \frac{1}{2}\left(1 + \frac{1}{3}\left(1 + \cdots \right) \right) \right) \\
\end{aligned}
\)

Python Program

Code:

s = 0
for k in range(13, 0, -1):
    s = 1 + s / k
    print(f"{s:>.9f}")

References

• e (mathematical constant)
  

---

I hope you realise in time when you have to stop.

[Thomas Klemm]

Natural Logarithm

Example

\(
\log(1.2) = \log(1 + 0.2) \approx 0.182321557
\)

Initialisation

DSP 9
-1
STO K
0.2
ENTER
ENTER
ENTER
11
STO M
÷

Loop

1
K
M+
÷
x<>y
-
×


Result

0.018181818
0.016363636
0.018949495
0.021210101
0.024329408
0.028467452
0.034306510
0.043138698
0.058038927
0.088392215
0.182321557


Explanation

\(
\begin{aligned}
\log(1+x)
&= x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}} - \cdots \\
&= x \cdot \left(\frac{1}{1} - x \cdot \left(\frac{1}{2} - x \cdot \left(\frac{1}{3} - \cdots \right) \right) \right)
\end{aligned}
\)

Python Program

Code:

x = 0.2
s = 0
for k in range(11, 0, -1):
    s = x * (1 / k - s)
    print(f"{s:>.9f}")

References

• Taylor series