# Binary system explained (part 1)

Hi,
in the last post I showed cool trick, how to use your fingers to calculate to 1,023. It was based on binary system but I did not explain it much there. Here I will go through the conversion from binary to decimal system, that we normally use, and back.

So all of these digits systems work with some number of symbols, 10 in decimal system for example. These digits means some number and when you run out of them you will just add another digit, for decimal system you have of course 0,1,2,3,4,5,6,7,8,9 and after 9 you have to start repeating previous symbols. Now if you think about it you could actually use this various ways, for example we write 10 but it could also be 00.

Binary system has 2 symbols only. 0 and 1. This means that you will have to use much more digits if you want to symbolise big number. Basically if you add one to number you display one higher symbol and you never skip any of them so there is given amount of numbers you can store in x number of digits. Because there are 10 symbols in decimal system you can display 100 different numbers (different combinations of these symbols) in two digits. From 0,1,2,3… to 97,98,99. This is 10² where the exponent is number of digits you have. For binary system the base will be two because you can store only two symbols in one digit. In two digits you can store 2² numbers.

This is good to know if we want to transfer from one system to the other, lets say that we have number in binary:

10011010

We want to change this number into decimal so that we understand it.
Every digit here stands for one of those exponents first digit (from the right) stands for 2º which is one and it either stores the number one or it does not. Since it is 0 and 0 is usually reffered to as empty, the first digit, again from the right, does not store any information[1].

2nd is for 2¹ and there is 1 which means that the information is there and we know that our final number (n) equals to the rest of the numbers plus 2.

We continue and there is no number on the next place but on the 3rd 2³ we have one which means that we remember this which equals 8. The next is 16 and the last one is even higher: 128. We now add all of them together to get 154 if I count right.

It is good to notice that with this system you can create any number you like because always this applies:

2ª=2^(a-1)+2^(a-2)+2^(a-3)..2^(1)+2^(0)+1

So
8=4+2+1+1

Because the post is quite long right now I will continue in the next post. I do not know when this will be because I am going on a trip this weekend and some days after but after I return there will definitely be new stuff 😉

Dragallur

[1]Same as zero in our system! If you have number: 00456 its just 456 of course that it is something different than 45600 where it moves the place where the numbers are but still it does not count to the final number!

# How to! 7) Count to 1,023 on your fingers

Hi,
as promised, weekend post is here! Ok, so I learned this cool thing when I was on seminar in Hamburg. First I thought that someone is pointing middle finger at me for fun (you will see the reason soon) but it was actually counting method. Though you have only 10 fingers you can use them to produce all numbers from zero to 1,023 which is cool.

If you pay close attention and you know something about computers you know that the number 1,023 is very special. It is 1,024 but one smaller (1,024 is actually the special number).

The thing is that 2^10 is 1,024. And in computers you work in binary system with only 0 or 1  ….    on or off and you get the number of combinations that you can arrange binary system if you put 2 to the power of digits you have. On fingers you can not arrange 1,024, you will see why[1]:

### Method

Turn you palms towards you. Since in Europe we write from left we will start with left thumb (palms still towards you). Now make fists.. that is number 0

Rise your thumb, that is 1. (1000000000)
Put only your index finger up, that is 2. (010000000)Put your thumb and index finger up, that is 3. (1100000000)  –> the number of digits shows the number of fingers you have.

So basically if you have number lets say 17. You want to transef it into binary. You will do this by subtracting the highest 2^x power which is equal or less to the number itself.

The 2^x numbers go like this: 1,2,4,8,16,32,64,128,256,512,1024…

In the case of 17 you will subtract 16 which is 5th number in the row. 5th finger on your palm is your left pinkie so you will put it up. Then you are left with 1 which you again subtract by the highest 2^x number which is equal or less and this time it is 1. 1 is first number and left thumb on your hands.

If you want 349 you have to subtract 256 (9th number – right index finger)
You are left with 93-64 (7th number – right ring finger)27-16 (5th number – left pinkie)
9-8 (4th number – left ring finger)
1-1 (1st number – left thumb)

Now train a little bit and impress your friends 😉 The key is to remember the first nine numbers and at what positions they are. Also you can show others just counting one by one. Just do not turn your hands towards them, numbers 4,5,128,640 and other could be dangerous 😀

Dragallur

[1]The reason why you can not count to 1,024 is of course that you are starting with 0 not 1 so your right thumb stands for 2^9 and not 2^10. You can produce another binary digit with your tongue 😉

# Still here, but in Germany

Hi
ok so I did not post for few days and I have an explanation ;). I started my exchange year in Germany. Actually right now I am third day in my host-family and second day in school.

Thinking about physics I did some pretty fun experiments with my host-brother though I have still lot to learn especially because it was electromagnetism which is not really my thing. I will have first class of physics the day after tomorrow but we have only 1.5 hours per week.

Today I had math in the local school. I will see how it goes because it seems that the class is really behind of what we were able to do in Czech Republic. I say able because it seemed to me that they had lack of some basics but they did successions.

Posts wont be posted very often I guess. I have lot of things to do but I will try to write something on the weekend. This also means that I wont be able to read your posts, if you want you can link me to some in the comments, that way I will take a look at it 😉

Dragallur

# Quick point about equations and graphs

Hi,
mliae asked me to make some simpler post so she understands this post. So here it comes, she said that it was long since she used equations:

Well equation is something like this:

x=1

That is quite clear. Of course you can have very complicated equation with many “unknowns” which are usually noted as letters, x for example. All thats easy and it says that something (x) is equal to one, it has the value of 1.

We can manipulate these equations if we abide one rule: both sides (from left and right of equal sign) have to be manipulated. If we add one we have to do it on both sides:

x+1=1+1     —>    x+1=2

Easy. We can substitute in equations if we work with more:

a=b+42
2a+3=x

2(b+42)+3=x    (“a” was substitued by b+42 because that is what it equals to.)

Now in the post that is our concern we used this equation:

To get here we have to use graph but I did not stop on that very much so I will go through it again.

Points on graph have two coordinates. This is because graph has 2 dimensions. These coordinates are usually called x and y and they are noted like this:

(x;y)

x says how much the point is to the left or right and y says how up or down (or closer/ further)

We had two points there on the graph:

Now we will take the blue point as stacionary of course but since we are working in general and not with specific numbers, it should not matter.

So black point has coordinates x and y (x;y)

Since we want it to be general we will left it like this except the y. This is because we are going to derive functions and in those y=f(x) which means that y coordinate of the point is f(x). That is the notation that is used. We are working with the function f that gave coordinate y to our black point.

black_point(x;f(x))
blue_point(x+h;f(x+h))

The h should be clear from last time. It is the distance from the black point. Then when you insert this into the “slope” equation which I talked about here, you will get what you want.

Dragallur

Feel free to ask for more clarification.

# Rosetta and OSIRIS-REx

Hi,
today, as promised I will look upon two missions that has to do a lot with small stuff flying around the Solar System.

Now I said stuff because Rosetta is a mission for comet and OSIRIS is mission for asteroid.

Rosetta is a mission that was launched back in 2004 by ESA which is European organization. It went for the comet 67P or also called Churyumov-Gerasimenko which kinda looks like duck:

Ok, fine, it does not but look here.. from this photo I would say that it is cat with huge tumor on back.

It went with Philae which is a lander module. It took 10 years to get there. It visited two other asteroids and went around Mars.

After some small changes it went to orbit around the comet even though it has escape velocity of 1 m/s.

Then it deployed Philae in 2014 but harpoons that should have eased the landing did not deploy and the site was much harder than it looked like before (the site was chosen because there was supposed to be “soft” regolith). It bounced twice and almost float away completely. It had battery for 2 days which were of course not enough to conduct all experiments and it could not recharge because it was under some cliff. Nobody knew where it was and we could not identify pictures that it took.

It puts me in awe to know that this picture is from a comet. (Philae sits in the right middle of the picture in shadow.)

Luckily Rosetta still orbiting the comet finally found it and put them all in context. The mission ends in 30th September and Rosetta will too crush on the surface.

The picture of Philae

Now that is for some asteroid exploration back in time.

Three days back, 8th September OSIRIS-REx, an asteroid study and sample return mission was launched.

The last part is pretty huge, yes USA is for the first time going to return samples from an asteroid to Earth (Utah is the landing site).

It launched on the often used Atlas V and the whole mission for asteorid called Bennu will take 7 years. OSIRIS will stay on its surface for whole 505 days! (Look how planned this whole thing is!)

There are lot of instruments on its board which I wont go through all. There are many cameras because OSIRIS will first orbit the asteroid and scan its surface to find a good place to land.

It has special leg that will try to take samples using gas of nitrogen. It can take up to 2 kilograms and enough nitrogen for three tries.

Dragallur

# XKCD

Hi,
again, it is kind of late so today I will only share with you some cool XKCD pictures and next post will probably be about our asteroid exploration which has 2 major news.

Not the best one but surely up-to-date.

Yeah cameras are great 😉

Thats how conspiracy is created.

Dragallur

All comics are from XKCD site, thanks Randall

# 2 awards in 1 post

Hi,
it seems that I was nominated for some awards, maybe there were more of them but I did not catch them, what a pity.

Liebster award from mathsbyagirl, thanks for this again 😉 It seems that I should answer those question too so here you go:

1. What is your passion? Astronomy
2. Why did you start blogging? Check this out.
3. If you could change one thing about yourself, what would it be? The thought that I want to change something?
4. Is there a moment in your life you wish you could go back to just to enjoy it again? If so, what moment is it? Sure there is, many. Anytime would be fine if I could live everything to this point again.
5. Are you planning to publish a book in the future? If so, what would it be about? Nope.
6. Do you believe in love at first sight? Not so much experience in this though it happens and it probably does not lead to best marriages, who knows, depends.
7. Is there something you’re afraid of that’s quite unusual? Not anything unusual probably.
8. What country are you from? Czech Republic
9. How many languages do you speak? Czech, English
10. What is one of your most embarrassing moments in life? Those in which I felt embarrassed but they did not need to be embarrassing at all.
11. Are you happy? Yup 😉

I guess that these answers also count for Versatile award by thenexusscience and it is getting late but if anybody wants to grab these awards go for it.

Dragallur

# Derivatives made easy 2) Non-differentiable functions

Hi,
so I continue with this tutorial. Today I will cover when function is not differentiable. If you want to read the basics check out the last post, it links to limits and continuity which will be important today.

Just to remind ourselves here is the formula that I explained last time:

It uses limit so logically if the point we are trying to derive is not continous, which means that the limit does not exist there we can not derive. This is not the only thing that can happen to us. Sometimes function is completely continous but from negative side it grows exponentially and from positive side it is linear. Check out the nice picture below:

From the whole picture the most important point is the second one. It says: “continous (no gap) but not smooth, not differentiable”. To find this out algebraically you need to know the equations of both left and right part of the function.[1]

If you know them you will insert it into the derivation equation that we used last time as f(x). You have to add h of course as it is the difference between your two points. If both of your solutions equal (right and left side) you know that the function IS differentiable.

Dragallur

PS: ask me if you want to expand any part of the post. I have no problem whatsoever with it and it will make me even happy to write some extra parts.

[1]Yes function can be made up of different parts only on intervals. You can even write something like (odd Xs are equal to 2).

# Derivatives made easy 1) Slope on curve

Hi,
today I am going to explain one fundamental equation that is used to calculate derivatives.

Derivative of a function basically says what is the rate of change or the slope of that function. You can have both rate of change of whole function which gives you another function or just at one point which equals to some number. The fundamental thing is to find the slope. Last time I explained this only for a line, not curve. On line the slope is constant but not on curve.

On the gif above you can see a function. There is a black point that is on the same place. Blue point is getting closer and closer with each step. All the time the black line creates secant line to the curve, it cuts it on two points. When it is red it is tangent line which means that it does not cuts it at all. It is the only one step in which it is just touching. To get the rate of change of curve you need to find this tangent line. Lets call the distance between the two points “h”. The tangent line will exist when h approaches zero.

THE equation. Sometimes f'(x) is written on the left.

This thing above is the equation that we will get when we take coordinates of both points.

black point(  x;f(x)   )                    [1]
blue point  (  x+h;f(x+h)  )           [1]

Just to clear things out, the first thing in the brackets is the x coordinate and the second y coordinate. As I said blue point is moved to side by the distance h and as the distance approaches 0 we are getting tangent line of curve. In the last post I showed that you need to only divide the difference of y coordinates by difference of x coordinates. Check it out if you do not know what I mean.

We will do exactly it to get right part of the equation above without the limit. Only h will be left in the denominator because there is the difference in x coordinates which means:

x+h-x=h

Now to get the tangent line we only add the limit and we have got the fundamental equation of derivates. This we can use for any curve or line that we want to derive. The left part is just a notation for derivation.

Dragallur

PS: be sure to check out this page that lets you interactively create secant and tangent lines.

[1]Just in case you do not understand why there is f(), it is basicly the same as y though we are in functions so we use this kind of notation and yes, f(x)=y. We just add h to get the y coordinate of blue point.

# 1st day of school + Formula 1 strategy

Hi,
today was the first day of school. Lot of people were pretty “stressed” though since I will be leaving to Germany in 14 days it is not so important to me. Already tomorrow we are going to learn normally. Yeah back in the same lines and system ;). At least there are some changes in our school, new computer class and some renovated library or what. Anyway I was reviewing some stuff from last year physics and found this cool stuff about Formulas.

When racing car drives, it is curve that slows it down most. To minimize this effect they have special tires and the following tactics:

When you are driving in curve your tires keep you from flying off because of friction. They act as centripetal force too. Huge role plays the size of the curve or its radius. So when the drivers want to turn right they need to move to the edge of the road and then smoothly turn exactly around the other rim of the circuit:

Great illustration of how the Formula drivers deal with curves, they use the “racing line”.

This way the centripetal force that you need is lower.

Dragallur

In the video below you can find sooner or later example of such tactics: