As someone who has owned a scientific calculator since
school, I am very aware on how I don’t need to use most of what one can do any
more – in fact, the simpler sums I need could often be done by my phone’s
calculator app instead. However, the vestigial memories of trigonometry and
matrices still in my mind from my Maths A-Level mean the need may come back, so
I may as well be ready.
The scientific calculator was invented by Hewlett-Packard in
1968, with the HP9100A, a cash register-sized machine with a TV screen, no
integrated circuits, and a five thousand-dollar price tag. Four years later,
the HP-35 was designed to co-founder Bill Hewlett’s brief to fit the 9100A into
his shirt pocket, becoming the first scientific calculator as we know it. In
1976, a later pocket model, the HP-65, became the first such calculator in
space, in case the guidance computer on the Apollo-Soyuz Test Project failed
(it didn’t).
With more programming ability than most will ever need, exploring
the functions of a scientific calculator can produce some interesting results.
On my model, I can move between decimal numerals (your usual 0-9, also known as
base 10) and other counting systems. For example, 666, becomes 29A in
hexadecimal (0-9 then A-F, base 16), and 1232 in octal (0-7, base 8) – therefore,
we have the Number of the Beast, the House Number of the Beast, and the PIN
Number of the Beast.
However, the most interesting function there is on my
calculator, and the one that produces many of the numbers we are asked to
observe in everyday life, is “log,” for logarithm. In the simplest terms,
logarithms are used to make long numbers easier to handle, by creating a
relationship between an exponent number, and the number that exponent needs to
be to raise the power of a base number to the value you want. You can make one
thousand by multiplying ten together three times (10 x 10 x 10 = 103
= 1000). Taking 10 at its base, the logarithm of 10 is 3, because you needed to
raise 10 three times (log10 1000 = 3) - likewise, the logarithm for
one million would be 6, and 26 would be about 1.415. Using the “log” button on
your calculator assumes base 10, the “common algorithm,” but base e (2.71828…),
a pi-like constant in mathematics, is also used, denoted by the “ln” key on your
calculator.
Logarithms are used to calculate earthquake magnitude
scales, the pH scale to measure the acidity or basicity of water-based
solutions, f-stops for camera shutter speeds, population growth, radioactive
decay, carbon dating, and decibels for measuring the intensity of sound. In
these examples, the figures on the scales presented to us in news stories, on
water bottles and on cameras are easily explainable scales, created from exponentially
growing numbers of abstract measurements – hydrogen ions per litre of water,
ratios of focal length to a camera’s entrance pupil, and so on, situations
where you just need a headline number to say, “oh, it’s that bit more.”
However, the headline figure should always be the starting
point for your understanding. You don’t necessarily need to know that a magnitude
2.0 hurricane is thirty-two times more powerful than one of magnitude 1.0,
multiplying by that amount with each full number, but knowing this brings
clarity to the pictures of destruction that accompany the figures. The same can
also be applied to interest rates on bank accounts and credit cards, and other
figures that may feel abstract, encouraging an easier relationship with
numbers.
That was always the intention with logarithms – they inspired
the invention of slide rules, using a mechanical slider on a printed scale to
make rapid multiplications and divisions, square roots, logarithms and more.
However, the ability to enter the numbers you need would wipe out slide rules
entirely, like when Bill Hewlett wanted the HP9100A shrunk into his shirt
pocket…
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