Numeral systems

today I will write a post about numeral systems and how they generally work. I already wrote about binary and decimal system, but in the last week I did some research on generally numeral systems.

So, just to recap. We normally use decimal system, this means that we use ten symbols to represent numbers, from 0 to 9. First place of number is the number itself, second position represents the number times 10 because we are in decimal system and 3rd represents number times 100 and so on. The general formula to transfer numbers to decimal system goes like this:

anbn + an − 1bn − 1 + an − 2bn − 2 + … + a0b0

a stands for the number that we see written, b stands for the system it is written in, n is the digit where a is –> 0 is the first place.

For example 1334 in the numeral system “7” would be:

4*1+3*7+3*49+1*343=515 in decimal system.

Note how you can represent easily any number in any system because of how the exponents rise in every single digit by one. If the numbers before the next digit are all full, like x9999 in decimal system or x4444 in base 5 system you always need to add only one to reach the next digit, this way there are no numbers left out.

What I really like is that if you had a machine that could change from any numeral system to other you could multiply numbers extremely easily. In decimal system if you multiply by 10 it is very easy because you just add one zero, why? Because all the numbers are “multiplied” by 10 with some exponent already and you just add one which moves all of the numbers to left. In binary system if you multiply by two you just move all the numbers by one digit:

101010111*10=1010101110      (10 is 2 in decimal) Or in 6 base system:
420351234*10=4203512340   (10 is 6 in decimal)

You are basically multiplying by “10” though you need to remember that the number is still in that system.

What about rational numbers? The stuff behind point?

I was really wondering about this and Wikipedia helped out! The exponents are simply negative as in the example below (binary number 10.11 to decimal system):

1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75

First place –> exponent 0
Second place –> exponent 1
First place behind point –> exponent -1Second place behind point –> exponent -2

Thats it for today.


Binary system explained (part 2)

in the last post I wrote about converting from binary to decimal number. Today I will continue, if you want to read the basics about binary just check out the post. This is also linked to my post about counting to 1,023 on your fingers.

So lets say that we have number 137 and we want to convert it into binary.

You have seen in the previous post that there is some highest digit that has the value of 1 instead of 0 which means that it stores the information[1]. We need to find out this value.

Its easy, its the highest 2ª number smaller or equal to our original value (137).

Such a number is 128 which is 2^7 so it is going to be the 8th number since we use 2º too (and 2º is on the 1st place).
Now we subtract it having 137-128=9 (1xxxxxxx)

Now we repeat with 9. The number that is smaller or equal is 2³=8
9-8=1 and the next 1 that stores information is on the fourth position. (10001xxx)

1 is easy because 2º also equals to 1 and it is on the first place. So 137 looks like this in binary: 10001001.

Lets try 759:

759-512=247 (10th number is the first 1) =1xxxxxxxxx
247-128=119 (8th number is 1, stores the information) =101xxxxxxx119-64=55 (7th number is 1) =1011xxxxxx
55-32=23 (6th number is 1) =10111xxxxx
23-16=7 (5th number is 1) =101111xxxx
7-4=3 (3rd number is 1) =10111101xx
3-2 … 1-1 === 1011110111 (quite lucky with so many ones ;))

Hope this all makes sense, if it does not just write in the comments below.

Btw. thought you have infinitely many systems that you can use, binary is the simplest of them all. You can not store information in less symbols because with one symbol you would not be able to distinquish where one information ends and another begins. You need to use “space” or some number or something.


Multiple star systems

Source: Wikipedia page Center of mass

Source: Wikipedia page Center of mass

today I will write about binary and multiple star systems. Those are systems where two or more stars are orbiting each other and sharing THE common center of mass.

Center of mass is exactly what it says. Center of mass of course can apply not only to stars but to humans, to planets and actually anything that has some mass.
On the picture you can see the estimated center of mass after gymnast performs cartwheel. (It is estimated because our body is so complicated that it is not easy to count where exactly it is.)

So this center of mass is a point in space towards which all the stuff is kinda turned.. so if you have a binary star system as is on the gif you can see that the smaller star moved the center of mass little bit towards itself and you can not actually say that the star orbits the larger one but both are orbiting the common center of mass or also the barycenter. Alpha, Beta and Proxima Centauri.jpg

For example Alpha, Beta and Proxima Centauri (circled) are trinary system. Or also Pluto and Charon are binary planet system.

So you can have binary system of two stars but also one of them can be neutron star, or even black hole.

Those multiple star systems are very common actually 1/3 of all stars are in multiple star systems. It does not end of course on binary or trinary systems but there are even septuple star systems like AR Cassiopeiae.

It is very difficult to calculate their barycenter so there is method with which you can simplify whole process a little bit. If there is more than two stars in the system you can make a hierarchy of all of them.

a) is not simplified at all
b) is clear binary system (nothing to simplify really)
c) is trinary system. Two closer stars were taken and their barycenter was calculated and then as if it was one star it was compared to the other star to create the final center of mass
d) is quadruple system where two barycenters each of two stars were combined
e,f) is now probably clear

In those systems planets can exist but their orbits have to be either small or huge compared to orbits of stars because otherwise they would be thrown out of the system. They can orbit only one or both stars.

One last thing. There are few ways by which we can detect them because usually we are File:Artist’s impression of eclipsing binary.oggnot able to distinquish star system and they look like one star to us.
First method is by eclipsing of one star behind the other. When one star crosses behind the other we can see the short dimness, this is similar to the method of finding exoplanets.

Astrometric binaries are binaries where one star seems to orbit around empty space which is usually neutron star, black hole or something which is not very bright.

Then sometimes when those stars are really close to each other one can transfer mass to the other one which can create accretion disk which we can observe.

Here are some other examples of binary star systems:


All of the pictures were taken from wikipedia pages Binary star, Star system, Center of mass and Alpha Centauri page as for information it was also taken from various wiki pages and also NASA (binary stars.)