February 28, 2020
Measuring takes an attribute from an object in nature and gives us a number
That is, measuring is a function from the real world into real numbers
Do not confuse measuring with calculating or evaluating
Can we measure 𝛑?
These three ideas are similar, but different
Since all intervals are comparable, we measure an attribute by comparing it with a reference
Civilized countries use the Système international d’unités (SI)
Normally, measurements are based in the physical property we care
In some cases, it may be easy to measure an electrical signal depending on the attribute to be measured
"A transducer is a device that converts energy from one form to another.
Usually a transducer converts a signal in one form of energy to a signal in another"
Photodiodes, phototransistors, photomultipliers – converts changing light levels into electrical signals
Photodetector or photoresistor or light dependent resistor (LDR) – converts changes in light levels into changes in electrical resistance
Thermistors – converts temperature into electrical resistance
Thermocouples – converts relative temperatures of metallic junctions to electrical voltage
Geiger-Müller tubes – converts incident ionizing radiation to an electrical impulse signal
Radio receivers converts electromagnetic transmissions to electrical signals.
My first smartphone was a Nokia N95. It had
So I tried to build a seismological station
In my first day of classes, the student union gave us a newspaper with this story in the first page:
The following concerns a question in a physics degree exam at the University of Copenhagen:
“How could you measure the height of a tall building, using a barometer?”
“You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof to the ground. The length of the string plus the length of the barometer will equal the height of the building.”
“Take the barometer up to the roof, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from the formula h = gt2/2”
“If the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow. Then you measure the length of the building’s shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height”
“But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in the gravitational restoring force T =2 𝜋 √(l /g).”
Tying a piece of string to the barometer, which is as long as the height of the building, and swinging it like a pendulum, and from the swing period, calculate the pendulum length.
“Or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up.”
“I would go to the building superintendent and offer him a brand-new barometer if he will tell me the height of the building!”
The expected answer: Measuring the pressure difference between ground and roof and calculating the height difference.
The story shows that the same attribute can be measured through different physical properties
Thinking “out of the box” enables measuring things that were not easy to measure before
Arithmetic coding is […] used in lossless data compression.
Normally, a string of characters […] is represented using a fixed number of bits per character, as in the ASCII code. […]
Arithmetic coding […] encodes the entire message into a single number, an arbitrary-precision fraction q where 0.0 ≤ q < 1.0.
 "All Wikipedia"
Same, encoded as numbers:
 33 76 76 0 55 73 75 73 80 69 68 73 65
Finally, in the form of a decimal number
An alien visited Earth, collected all observations in a big file—let’s say, all Wikipedia
Then the alien encoded it using arithmetic code, resulting in a value q
Finally, the alien took a 1 meter bar, and made a mark at q meters
The bar is sent back to the alien country
What is the problem with this idea?
Our tools have limited “resolution”
We cannot “resolve” the difference between small variations
With higher precision we get a new problem:
Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results
In this context, people has defined the following ideas
BS ISO 5725-1: “Accuracy (trueness and precision) of measurement methods and results - Part 1: General principles and definitions.”, p.1 (1994)
In some old material, people say “accuracy” in place of trueness
Other people say bias
These words are still common in science and technology
Be aware of this discrepancy
Measure twice, cut once
Measure once, cut twice
The uncertainty of a single measurement is limited by the precision and accuracy of the measuring instrument, along with any other factors that might affect the ability of the experimenter to make the measurement.
to measure the diameter of a tennis ball
The limiting factor with the meter stick is parallax, while the second case is limited by ambiguity in the definition of the tennis ball’s diameter (it’s fuzzy!)
In both of these cases, the uncertainty is greater than the smallest divisions marked on the measuring tool (likely 1 mm and 0.05 mm respectively)
Unfortunately, there is no general rule for determining the uncertainty in all measurements.
The experimenter is the one who can best evaluate and quantify the uncertainty of a measurement based on all the possible factors that affect the result.
Therefore, the person making the measurement has the obligation to make the best judgment possible and report the uncertainty in a way that clearly explains what the uncertainty represents
Measure the length and width of the table
Read the short story “Funes the Memorious” (by Argentine writer Jorge Luis Borges). You can find it on the web.