Psudo random numbers come from a special set of mathematical equations which act as the basis for natural processes. These are known as nonlinear dynamic equations.

Their outputs feed back into their inputs. They show areas of high initial sensitivity where any tiny change in input totally changes the output over time. Finally, they often show areas of different cycling behavior. The branch of math which studies them is homomorphic dynamics.

The psudo-randomness of slightly different seed values generating wildly different outputs has to do sensitivity to initial conditions. This is a property of the underlying paramater space structures those flat number line seqiences of psudorandom walk cycles embed themselves into in higher dimensions of the complex plane.

Graphing and iterating psudo random equations in the conplex plane shows infinitely complex geometric structures called julia sets which govern how algebraic numbers cycle through pseudorandom walks depending on initial seed values and equation used. These julia sets often are fractals with infinite complexity at its borders at all scales of precision. Fractal geometry is a fascinating can of worms.

Simply put, you put a random number in and it spits a more-or-less random number out.

A big issue with psudo randoms is they will always give the same series numbers if you begin the equation with the same computationally finite seed values. You could the generated sequence of numbers to work back and find the seed values and equation used to generate them. This is a serious security concern when using them for cryptography. The amount of computational work it takes to work back is massive but its doable with modern quantum super computers.

The mechanics of pseudo random numbers comes from statistical combinatorics, nonlinear algebra,fractals, chaos theory, and sensitivity to initial conditions.

True random numbers come from directly measuring physical phenomenon with sufficient randomness in their mechanics.

Things like the decay of a radioactive isotope or lava lamp turbulence have built in randomness. There is no seed or way to generate the same sequence of motions or predicting when isotopes decay.

Turbulance for example has fractal properties in its energy distribution as well as random brownian motion adding up on the atomic scale. Radioactive half life has uncertainty principal built into it. These universal operations have true uncalculatable randomness thanks to entropy, the uncertainty principle, fractals, brownian motion, chaos theory, and sensitivitiy to initial conditions.

The physical universe is the most powerful computer there will ever be. It calculates with infinite decimal precision in its mixed mathematical, statistical, and physical operations. It uses real trancendental like pi numbers with infinite non-repeating decimals, and does its calculations at the speed of causality/light.

Our best super computers will never be infinitely powerful. Our numbers need to be finite and computable to work with them and understand them. The universe could not care less if its values are finitely computable or usable for human work.

So theres fundamental limits to how random we can get through artificial computer algorithm generation using computable numbers. True randomness through physical processes leverages the universes in built infinite precision and mechanical algorithms as a black box and just measures the output result.

Cloudflare uses a wall of lava lamps for some of their RNG needs

Math! Also, noise!

There are algorithms (a set of math steps) that make pseudo-random numbers. These usually involve large prime numbers, because those usually generate fewer repeating patterns.

A truly random number generator is similar to rolling dice: you use some source of randomness and convert it to a number. All electric circuits produce “noise” (which is often received radio waves and such that interfere with the circuits). Think of tuning a radio to a channel with nothing on it–you get “white noise”, which can be a good source of random information. Then all you need to do is convert that to a range of numbers, and you’re good to go.

These are fairly simplified explanations, so take them with a grain of salt, but they give the general idea.

I see a lot of good answers here but let’s try it from another angle.

How do we get randomness from a function or formula?

For starters let’s setup a few simple rules.

Every time our random function is called we’ll

• Take the last output from a variable we call LAST_RESULT
• If there’s no value in LAST_RESULT we’ll assume the value is 1
• We run a set of calculations storing the value in a variable we call X
• We store the result of these calculations in LAST_RESULT
• We return this new “random” number.

So let’s call it.

> Random()
Since LAST_RESULT is undefined SET LAST_RESULT to the value of 1
Set X to the result of this calculation 
   (LAST_RESULT+1) * 3

X is now 6

Set X to the result of this calculation
   (X + 7) / 2

X is now 7

Set X to the result of this calculation (rounding to the nearest whole number)
   X/LAST_RESULT

X is now 7

Set LAST_RESULT to the value of X

LAST_RESULT is now 7

Return the value of X as the result 

Result is 7

Ok. So let’s call it again

 > Random()
Set X to the result of this calculation 
   (LAST_RESULT+1) * 3

X is now 24

Set X to the result of this calculation
   (X + 7) / 2

X is now 16

Set X to the result of this calculation (rounding to the nearest whole number)
   X/LAST_RESULT

X is now 2

Set LAST_RESULT to the value of X

LAST_RESULT is now 2

Return the value of X as the result

Result is 2

And if we call it again we get seemingly random results

Random() Result is 4

Random() Result is 3

But the next time you run it you’ll get the same results in the same order. 7, then 2 then 4 then 3

So what you need is something to “seed” the random number calculation.

Something like

SetRandomSeed Set LAST_RESULT to the current second of the day

Then when you call Random after this it starts with that as the prior results and gives seemingly random results.

Of course my calculations are rough and probably fail/repeat after so many calls but it gives you an idea of how this works.

So the trick is to get noise for the seed. That could be the number of non leap seconds since 00:00:00 UTC on Thursday, 1 January 1970 (Unix epoch)

Or the temperature reading of a CPU chip.

Maybe it’s the ratio of red vs yellow from a camera feed looking at lava lamps.

Or the current users average typing speed.

An additional note. Many of those would not be “cryptographically” secure for encryption because they can easily be determined by a third party. We all experience the same “Unix epoch” within a few milliseconds if our system clocks are properly set for example. Or monitored from afar and reproduced (hacked webcam shows they had just typed the following letters in the previous 27 seconds that we know the “algorithm” uses, etc.

From my opinion it is more computer science sorcery than math sorcery.

For true random generation you usually need some specialized hardware for it, that uses sone natural source of random. One could use the decay of a radioactive material as such a source or the noise one can get from audio input. Unfortunately, I don’t know what actual hardware uses.

For pseudo random generation, you usually use a seed (ideally a true random value or something with a high entropy) which you feed into an algorithm like Linear Congruental Generator (LCG) or Mersenne Twister (there are lots of algorithms).

One further important note: Tge use case forvwhich you need random numbers is important. A video game could accept a random number generator with “lower” quality while a cryptographic algorithm always needs a cryptographic secure random number generator (don’t forget: “don’t roll your own crypto”).

Finally there are quasi randim number generators, however this name is very misleading. The mathematical correct term is low discrepancy sequence. There are not random at all but can be used and have useful properties in some settungs where pseudo random number generators can be used. Never in a cryptographic algorithm, though.

They get some dude hammered drunk, then stand him by a pole and ask him to take 100 steps.

You can use physical objects like dice or lava lamps that will naturally form random distribution when we check. But Newton and others would argue that even this was a determinant problem and if you had perfect knowledge of the dice and a good physics theory, you could predict the outcome.

We can only recognize randomness by the patterns it leaves behind.

The philosophical truth is that we don’t know if “randomness” is an actual phenomena or just a bucket where we put outcomes we haven’t learned to predict yet. A sort of randomness of the gap. Some have suggested that as a pattern-recognizing machine, the human mind simply can’t conceive randomness. Even the way “randomness” is verified is by looking at the distribution in the outcome and see if it matches the pattern we expect.

PRNGs aren’t random at all; they produce a deterministic sequence of numbers based on a seed value and an internal counter. Two PRNGs using the same algorithm and seed will produce the same sequence of numbers. The sequence is difficult to predict without knowing the algorithm and seed, and the values are close to evenly-distributed, which is enough like random numbers for a lot of use cases.

Here’s an example in Ruby:

seed = Random.new_seed()
=> 142757148148443078663499575299582907518
prng_1 = Random.new(seed=seed)
prng_1.rand()
=> 0.6702742156250219
prng_2 = Random.new(seed=seed)
prng_2.rand()
=> 0.6702742156250219
prng_1.rand()
=> 0.9667236181962573
prng_2.rand()
=> 0.9667236181962573

If you run this yourself using 142757148148443078663499575299582907518 as the seed, your first two pseudorandom numbers will also be 0.6702742156250219 and 0.9667236181962573, assuming your version of Ruby hasn’t changed its PRNG.

All you need is a bunch of lava lamps and a camera.

Nice try skynet