Stumbling Intuition #2: 0.999999 (Ad Infinitum)

[Debater Slayer]

As the second entry in a series of threads partially meant to show how intuition can sometimes fail to give correct answers in math and science, I'm going to talk about the recurring decimal 0.9999..., which, in the real number system (the one we most typically use), equals 1:

0.999... - Wikipedia...

There are multiple proofs of this, ranging from simple ones to others employing advanced mathematics. Consider the fraction 1/3, for example: in decimal form, this is 0.3333... with recurrence of 3 ad infinitum. Now also consider that 1/3 multiplied by 3 equals 1. This means 0.3333... multiplied by 3, which equals 0.99999..., is also 1.

This is a simple proof of the above equality, courtesy of the Japanese Wikipedia entry on the recurring decimal 0.9999...:

https://ja.wikipedia.org/wiki/0.999...

There are other proofs using series, limits, and various other mathematical concepts.

So, while it may not be immediately intuitive to think of the whole number 1 as an infinitely recurring decimal or as a series summing to 1, that's exactly what it is in the number system we often use.

(This doesn't mean I'm 0.9999... people, though... or does it? Eek!)

 

[Revoltingest]

It requires accepting the meaning of infinity.
Then all is clear.

 

[VoidCat]

My mind broke just now. Don't know how to fix it. Still it is nice reading you geek out over the math stuff. I legit can imagine you smiling and talking with a invested and passionate tone when sharing said info with folk in person and that for some reason is comforting when reading your post it gives that same energy of being very passionate about something and i love it. Kinda reminds me of how i act when talking about autism or stuffed animals.

 

[VoidCat]

. Consider the fraction 1/3, for example: in decimal form, this is 0.3333... with recurrence of 3 ad infinitum. Now also consider that 1/3 multiplied by 3 equals 1. This means 0.3333... multiplied by 3, which equals 0.99999..., is also 1.

I think i get it now rereading it. This kinda makes sense

 

[The Hammer]

I'm 99.999999% convinced math is made up on the fly, like language is.

 

[Shadow Wolf]

As the second entry in a series of threads partially meant to show how intuition can sometimes fail to give correct answers in math and science, I'm going to talk about the recurring decimal 0.9999..., which, in the real number system (the one we most typically use), equals 1:

0.999... - Wikipedia...

There are multiple proofs of this, ranging from simple ones to others employing advanced mathematics. Consider the fraction 1/3, for example: in decimal form, this is 0.3333... with recurrence of 3 ad infinitum. Now also consider that 1/3 multiplied by 3 equals 1. This means 0.3333... multiplied by 3, which equals 0.99999..., is also 1.

This is a simple proof of the above equality, courtesy of the Japanese Wikipedia entry on the recurring decimal 0.9999...:

https://ja.wikipedia.org/wiki/0.999...

There are other proofs using series, limits, and various other mathematical concepts.

So, while it may not be immediately intuitive to think of the whole number 1 as an infinitely recurring decimal or as a series summing to 1, that's exactly what it is in the number system we often use.

(This doesn't mean I'm 0.9999... people, though... or does it? Eek!)

I learned that either in high school or early in college.

 

[Debater Slayer]

I'm 99.999999% convinced math is made up on the fly, like language is.

The issue of whether math is created or discovered is quite complicated, but there are many mathematical relations that mathematicians and scientists simply reach through research and logic rather than invention. Mathematics is the language through which we, as humans, understand a lot of the universe's laws, but that doesn't mean it's entirely invented rather than discovered.

Some mathematical constants frequently appear in the sciences and in solutions to mathematical problems, as if they were inherently woven into the fabric of natural law. We could try altering the numbers in a solution to, say, an equation of a projectile, but that kind of invention sometimes wouldn't work in practice if the goal were to accurately predict natural phenomena: our predictions about the motion of the projectile would fail simply because the numbers wouldn't match observations, data, and experiments involving the motion of a projectile.

This may be germane here:

A constant is only fundamental by convention. For our adjustments, obvious constants are those that appear in basic physical and chemical theory. The Planck constant h, the speed of light in vacuum c, and the elementary charge e are basic quantities in quantum mechanics, general relativity, and quantum electrodynamics. They are exactly defined in the International System of Units (SI). Avogadro’s constant NA and the Boltzmann constant k are also exact in the SI. The former sets the number of particles in a mole, while the latter connects the Kelvin temperature scale with energies in statistical mechanics and thermodynamics.

Important inexact constants are the Newtonian constant of gravitation G and the dimensionless fine-structure constant α. Masses and magnetic moments of the lightest charged leptons, i.e. the electron and muon, and of light nuclei with charge number Z=0, 1, and 2 also fall within the scope of our work as their precise evaluation often involves knowledge of the fine-structure constant, etc. We provide masses in the SI unit kg and as well as relative atomic masses in the atomic mass unit u (or mu). The Newtonian constant of gravitation and fine-structure constant are currently known to five and ten significant digits, respectively. The g factor of the free electron and the Rydberg constant are even better known.

Fundamental Constants in Nature

Of course, it is entirely possible that some of the quantities we currently consider to be constants, per our existing models, may change or be further revised in the future, as has happened numerous times throughout history. But then that wouldn't necessarily be an act of invention on the part of mathematicians or scientists; it could merely be the outcome of following scientific and mathematical models and getting specific results from them.