Is 0.999... = 1? (spoiler alert: no it is not)

You may have encountered the popular claim that \( 0.999... = 1 \), where the three dots signify that the decimal continues forever. This is a somewhat weird claim, since it would mean that mathematics is broken. There should be no way for two different numbers to have the same value. What makes it weirder is that this is quite popular claim. I've even seen mathematicians say that it's true! But is it though? One popular proof is to first denote \( S = 0.999...\) and then multiply by \(10\) to get \( 10S = 9.999...\) and subtract \( S \) from it, to get  \( 10S - S = 9.000...\) and finally dividing by \(9\) yields  \( S = 1.000... = 1 \) and we see that  \(0.999... = 1\)! However, there's a problem. This short derivation is not strictly speaking correct. It is veeeery close to being correct, and to see why let's look at finite decimals first. Let's say that \(S = 0.999\) (note that this is not the same as \(S = 0.999...\) ). Let's do the same trick as before, so

Le Grand K announces retirement, at last

Starting from 1889 and scheduled to become emeritus (emerita?) sometime in 2019, the International Prototype Kilogram (also known as Le Grand K) had a service of 130 years. Which is a remarkably long time for a standard of measurement based on a metallic cylinder.


The change is rather massive (pun intended), because it affects every SI-system unit. However, the biggest change is not the kilogram, but the fact that some of the old SI units had a dependence on measured values. The new definition relates the base units to constants of nature, which will be redefined as being exact.

The speed of light is a good example of setting some constant to be exact. We used to measure it's value, which leads to experimental error. But then in the 15th General Conference on Weights and Measures in 1975 it was decided to set it at exactly 299 792 458 meters per second. This also allowed us to define the meter in a very precise way, by establishing some temporal yardstick.

Similarly, to get a precise kilogram, some ground work had to be laid out first. To begin with, we needed a very precise way of measuring the kilogram that is somehow related to the constants of nature, and the Kibble balance is one such method. The device works by balancing a massive object against a magnetic field, and it is an extremely precise way to measure mass. It's also an extremely difficult and expensive way of measuring mass.

Kibble balance experiments have been going on for at least 20 something years, and scientists are just now beginning to realize the measurement with sufficient accuracy and precision for the SI redefinition. And to top it all off, there were several competing measurements aiming to do the same thing, so time was of the essence. Probably the most notable alternative is the Avogadro project, which aims to make a single crystalline ball of silicone that will act as a weight prototype.

But isn't this just Le Grand K all over again? Not quite, because they would also redefine the kilogram as the weight of exactly 35.743734901285 moles of\(~^{28}\)Si atoms. Yeap, they are trying to measure the exact number of atoms in their prototype of kilogram, which is in the neighborhood of \(2 \times 10^{28}\) atoms overall. I would not want to be in charge of the counting!

This approach has the advantage that if we were to set the atomic mass of silicon to be exact (together with the Avogadro's constant), we would get an exact kilogram as well. However, this definition suffers from similar problems as the old kilogram prototype: we can never be exactly sure what contaminants it has, and thus it is no longer exactly right. And setting some atomic mass constant is a bit iffy, not to mention that you probably can't measure the number of atoms in a ball like that with perfect accuracy.

That is why the electronic Kibble balance has gained such popularity. But to get it working as well as possible, we also have to fix some constants and definitions. This means that values such as Planck's constant and electrical charge need to be exact. Additionally, the definition of ampere needs to be revamped. The old definition was based on a force between two infinitely long conductors one meter apart, and it never really made any sense to me. The new, much better definition, is based on the number of electrical charges passing through a surface per second, i.e. charge flux through a conductor.

Armed with these, it is possible to measure the weights of objects very precisely and it is no longer necessary to define the kilogram with some physical object. You can just decide that one kilogram is exactly the weight that corresponds to some fixed amperage in the Kibble balance and BAM! You have a fixed reference that can be applied to any weight.

All technicalities aside, I for one welcome this redefinition of the SI system with open arms. We finally have a set of base units that do not depend on any physical objects, meaning that they are the same for everyone! This is by far the most important change to the measurement system, although it will not affect our daily lives in any way.

Comments

Popular posts from this blog

Is 0.999... = 1? (spoiler alert: no it is not)

The Nobel prize in physics 2018: light all the way

Is it possible to make a laser out of wood?