Featured

## Solving Flows in Network Formats

Flows in Networks

Flows in Networks are used in problems such as traffic on roads and oil in pipelines. This graph is called a digraph where the arcs have a direction denoted by an arrow. The weights on the arcs are the capacities of the arc and show the limit of the flow that can be passes down that arc.

Consider a network with N nodes. Node 1 will be the source (all flow will come from node 1). Node N will be the sink (all flow will be to node N). For all the other nodes we will assume conservation of flow, i.e. no flow is lost.

The object is to maximise the total flow from node 1 to node N.

Intuitive Approach Step 1. Find any flow-augmenting path from source to sink such that each of the arcs has a positive flow capacity.

Step 2. The arc with the smallest flow capacity limits the flow along the path. Assign (i.e. ‘send’) this flow (f) and reduce the capacities of the arcs on this path by f.

Step 3. Repeat steps 1 and 2 until no suitable path can be found.

e.g. Initial Flow-augmenting Path 1 ® 2 ® 4 ® 6 f1 = 4 Second Flow-augmenting Path 1 ® 3 ® 5 ® 6 f2 = 4

Third Flow-augmenting Path 1® 3 ® 4 ® 6 f3 = 1 Maximal flow = 4 + 4 + 1 = 9 (this is obviously maximal since all the arcs from the source are saturated).

It’s often difficult to see useful applications for these diagrams but they’re actually very useful in real life.   For instance creating such structure would have definitely helped programmers and designers who created the Netflix algorithm which creates your suggested option screens.  Using the data from previous selections it can use flow format theorems to create a likely pattern of other shows.  Just look carefully how the structure works, if you haven’t got Netflix access then just use this trial to watch the US version!

If however our initial flow-augmenting path was 1® 3 ® 4 ® 6, f1 = 5, then the network would be stuck but not at maximal flow. We must therefore build in a refinement to stop us getting stuck. The idea is to allow a ‘fictitious’ back flow in the wrong direction along an arc in order to cancel out all, or part, of a previously assigned flow. e.g. in the above stuck network we could send up to 5 from node 4 to node 3 and still end up with a real flow from node 3 to node 4. So the second path is 1 ® 2 ® 4 ® 3 ® 5® 6, f2 = 4. Now continuing as before we could again have a flow of 9. (The real flow along (3,4) is 1 from node 3 to node 4).

A different notation can be used called the labelling procedure and appears to be preferred by the examination board.  They are commonly used in network diagrams too if you want to illustrate the flow from something like a e-commerce server to an ATC proxy, or other network devices. In this method when the capacity is reduced after flow has been assigned a back flow in the opposite direction is shown. This can help in two ways; one is by showing the total flow after a few iterations along a certain arc, even if it is the “wrong” direction. The second is it may help with the fictitious flows mentioned above.

## Watch BBC iPlayer by Proxy USA

Not strictly a maths post, but more a need in order for me to watch a programme being rebroadcast and thus watch BBC Iplayer in USA.  It was released a year or so ago, but I have been tipped off that it is being shown again and so for US citizens possibly a chance to catch it on the IPlayer application.  The series is called The Story of Maths and is presented by Marcus du Sautoy, it chronicles the history of maths through the ages and I am expecting great things.

But first the problem – BBC Iplayer doesn’t work in the US as when you try and access it does some sort of lookup and blocks non-UK addresses.  So I did some reading on geotargeting, proxies, VPNs and IP addresses and finally found a way to bypass these blocks and watch whatever I like.

So here it is –

# How to Access :BBC iPlayer for America by Proxy

Now I don’t know if you’ve tried at all but when you visit the BBC Iplayer abroad, if you’re not in the United Kingdom, you get redirected to the Radio section (which you can use).  However if you persist and try and watch something on the main BBC site – you get the following message below. This happens whatever you try to watch and it’s purely down to the location registered to your IP address.  At this point of course many people give up or try but it’s actually not that difficult to bypass these blocks.

Firstly you need to know what’s happening – when you connect to the BBC website your IP address is recorded.  This is the unique number that is assigned to you by your ISP when you connect to the internet, everyone has one.  Next this address is checked against a large database which contains all assigned IP address ranges and which country they’ve been allocated to.  So if you’re in the US or using a USA Proxy then you’ll have an American assigned address, explained here.

That’s pretty much it, if your IP address is registered anywhere other than the UK then you’ll get the above message. So in theory all you have to do is change your IP address so it’s a British one and not American registered.

Now you can’t change the initial address you’re allocated because a US ISP can only assign the addresses it has been allocated.  Occasionally you might get lucky and an address range is incorrectly listed in  the database – but that’s a bit of a long shot.

The trick is to use a proxy or VPN service to connect through.  This is a server that sits between you and the website you visit, it simply forwards all information acting like a middle man.  However if this server is based in the UK, the BBC application will think you are also based in the United Kingdom.

So here’s the application I used, there are lots of others available but I used this one because it’s one of the cheapest and I know someone who worked on some of the code so I know it’s a legitimate company.   Here’s the screenshot from the program I use called Identity Cloaker. This is the screen where I select which server to connect to, they have lots of countries but I need a UK one. After clicking on a UK server I then can go back to the BBC Iplayer site and try again.  This time I connect the site sees a UK based IP address (from the proxy) and so allows me to watch whatever I want. In fact you can even disconnect the proxy after the program has started and it will still work as the IP address is only checked at the beginning of the show. So that’s it really – how you can access BBC Iplayer in America! I’m really pleased with Identity Cloaker,  it’s super fast and very easy to use – you can get a 10 day trial here if you want to try it.  My only issue is that there’s no sign of The Story of Maths yet so I have to keep checking!

http://www.theninjaproxy.org/tv/how-to-use-a-bbc-iplayer-proxy/

## Introducing the Invariance Theorem

This is from a series of lectures by by Hector Zenil and Narsis A. Kianib entitled From Networks to Cells. It is sponsored by the Sante Fe Institute.

Text Transcription below:

So we saw in the last lectures how we can quantify algorithmic randomness by using computer programs running on some reference universal Turing machine. But one could think that it is always possible to find some language in which a particular object had a short encoding, no matter how random. For example, Alice and Bob could agree to compress all the content of the Encyclopaedia Britannica in a few symbols, so when Alice presents Bob with those symbols, Bob would know Alice meant the Encyclopaedia Britannica.

Algorithmic Information Theory (AIT) would mean little if the complexity of something could be determined so arbitrarily by renaming things with any number of symbols. AIT requires the program to reconstruct the original object from scratch so there is no coding cheating. Bob would need a computer program to reconstruct the Encyclopaedia Britannica without any external help from Alice or anyone else. It would also look like one would need to specify the specific programming language and the particular universal Turing machine on which those computer programs would run for algorithmic complexity to work or make sense. Otherwise, one would be able “rename” things and make anything look random or simple by changing the underlying language and Turing machine. So let’s consider the following sequence that has 3 symbols, 0, 1, and 2: We can show that this sequence can be generated by a small computer program like this one: whose length is no more than 56 bytes no matter how long the sequence to produce the same pattern. However that is in the Wolfram Language running on Mathematica. What if instead we had used Lisp, Java or Visual basic? It seems that the result would then depend on the computer language chosen. And even in a single computer language there may be different computer programs of different size can produce the same object. For example, the following two computer programs can generate any number of digits of the Thue-Morse sequence and both are different in length even though both are small and generate the same sequence. So how to deal with different computer programs in possibly different computer languages with different lengths? Fortunately, the Invariance Theorem as proven by Kolmogorov, Solomonoff and Chaitin, tells us that the difference between any two computer programs producing the same object is at most a constant from each other. More formally, the so-called invariance theorem establishes that the difference of the lengths of the minimal programs on any two computer languages L1 and L2 is always bounded by a constant that depends on L1 and L2 but not of s: The way to see this is by thinking in a translator program between two languages L1 and L2.

Because what the invariance theorem says is that one can always write a 3rd computer program of fixed length capable of translating between any two languages L1 and L2. This is actually very clear with the computer language Java because one of the innovations of Java when it was introduced was that there was a Java Virtual Machine or JVM that made Java multiplatform in the sense that you could run your Java program on any platform without changing any language code. That is because the many JVMs were acting as a translator, or officially called a compiler, between the Java programming language and each machine code for different platforms such as Unix and Windows. So if the translator is shown in blue, one has to only add the translator to every computer program written in some language to convert it into another program in another language, both languages producing the same sequence of interest.

The length of the compiler or the length of the blue part is the same for all input strings and so it is constant. So if L1 is the Wolfram Language and L2 is Java, then the invariance theorem tells that finding the shortest computer programs to measure the algorithmic complexity of a string is about the same length up to a constant. In another more practical example, what the “c” in the invariance theorem characterises in a biological context, for example, is that the DNA and a set of reading and writing chemical machines called ribosomes form a couple of language and compiler able to transcribe and translate DNA into proteins effectively from one language into another. So ribosomes can be thought as some sort of compilers between the DNA, RNA and the protein space with the ribosomes always of same complexity independent of what they are reading or writing. can be reproduced with a code no longer than 56 bytes no matter how long the sequence with this pattern is required.

Well, that is not entirely true. Actually the code has to grow by a little bit if we want to produce longer sequences with the same pattern. It does grow a little bit because one has to determine the end of the algorithm, that is when do we wish the sequence to come to an end. And that number does make the code to grow a bit. In fact, classical information theory tells us that we can encode an integer n in about logarithm of n bits, because remember we can always guess a number in about logarithm of n yes and no answers. You can see how running this code with very large numbers, the byte count does actually change for very large numbers: So despite how trivial this extremely little piece of information may look and how kind of little difference it makes, it is not only mathematically precise but it will turn out to be the most important little piece of information in the field of Algorithmic Information Dynamics, and you will see soon why.

Not to keep you waiting but not getting into details, the main idea is that, unlike a deterministic system that has a well-defined generating mechanism such as this sequence and its generating formula, other systems that may be subject to some source of randomness or are interacting with other systems, this logarithmic change will not be longer be respected and the more removed from the logarithmic term the more it will tell you some unvaluable about the system’s behaviour and its underlying causes. To understand this better, in the next unit we will cover the theory of dynamical systems for you to understand what a dynamical system is before we finally get into the subject of Algorithmic Information Dynamics. .

Some great Maths Lecture on British TV – to access them from outside the UK then you’ll need a VPN. This post shows how it’s done – ITV Hub abroad.

To test functions of random generation in tandem with online programs and devices then you’ll probably need to switch IP addresses randomly while conduction the tests by perhaps using rotating proxies – https://residentialip.net/what-are-rotating-proxies-and-how-can-they-make-me-rich/

## A Basic Introduction to Statistics

So how can we define “statistics”?  If you search around the internet you’ll find lots of examples, but it’s a useful exercise to try and  use your own words.   How about this one –  statistics the “science” of obtaining information from numerical data –

• This information is used to
• gain insights (Who will be the next president?)
• understand relationships (Does watching T.V. affect a student’s grades?)
• draw conclusions (Does smoking cause cancer?)
• make decisions (Should the space shuttle Challenger be launched?)

Challenger was launched in 1985, even though there were concerns that the temperature was too cold.
The space shuttle exploded a few minutes after take-off.
Three Main Components of Statistics

• Data collection
• Data analysis
• Drawing conclusions from the data

What is a part of components? Statistics is driven by data, therefore …
“In God we trust. All others must bring data.”
Robert Hayden, Plymouth State College

Statistics is more of an art than a science.

We can never determine if an event will definitely occur: we can only determine the probability that an
event will occur.

“Statistics means never having to say you are certain”

This seems to be a major shortcoming, but it is actually the greatest strength of statistics!

“An approximate answer to the right question is worth a good deal more than the exact

John Tukey

In mathematics we frequently find exact answers to fictitious problems. Statistics gives approximate
answers to real problems. The difficulty is asking the right questions and collecting the correct data!

A mathematician and a statistician apply for the same job. At the interview, they are
asked the question, “What is 1 + I?” The mathematician says “2” without hesitating.
The statistician pauses for a few minutes and asks “What do you want it to be?”

In statistics, different conclusions are possible depending on which questions are asked and which data
are collected.  For example if you’re studying the relative benefits of how to buy and resell sneakers online it’s important to gather the correct benefits too.  This is essential to study profits and whether your efforts are worth it.  Many entrepreneurs are faced with the issue of trying to balance the costs of buying – e.g marketing against the potential sales they might achieve.  There are other costs for smaller businesses, things like software and even specialised servers known as ‘sneaker proxies’ – read about it here.

Individuals are often accused of using statistics to distort the truth.
“There are three kinds of lies: lies, damn lies, and statistics.”
Benjamin Disraeli
A Lie is bad, a damn lie is worse, but a lie based on statistics is the worst lie! Why is a fie based on statistics
the worst type of lie? People believe it ;s true!
“A statistician is a person who comes to the rescue of figures that cannot lie for
themselves. ”

Statistics can be used to spin data. Forexample, a textbook publisher claims that a new math textbook
will increase pass rates on Regents exams. Data are collected which show that 2 of 30 students passed
the Regents exam using the old book while 3 of 30 students passed using the new book. Theclaim? The
new book increased the pass rates by 50%!
But do not despair!
“It is easy to lie with statistics, but it is easier to lie without them.”
Frederick Mosteller
However, always remember that 39% of data is made up. (Thinkabout it!)

## The Route Inspection Problem (Chinese Postman)

The Route Inspection Problem

The problem is to start from a node, travel along each and every arc and return to the starting point.  If it is possible to achieve this without going over the same arc twice, the minimum distance is just the sum of the arc lengths.  If it is necessary to cover some of the arcs twice, them we must select these in the most economic way.   This type of problem is called a Route Inspection Problem.  They are usually illustrated by gritter lorries or postmen, i.e. where all roads must be covered.

First we need to cover some network theory.

Node type – a n-node is a node where n arcs join 1-node, 3-node, … are called odd nodes

2-node, 4-node, … are called even nodes

In a network there must always be an even number of odd nodes.

Traversability – a network is said to be traversable if you can draw it without removing your pen from the paper and without retracing the same arc twice.

Looking at this network you will find you must start at one of the odd nodes and end at the other.

If you investigate other networks and note their node type numbers for each number you will find : For a network to be traversable it must have 0 or 2 odd nodes, and if we are to be able to start and finish at the same node it must have no odd nodes.  The implications of this result about traversability for route inspection problems are as follows.

If there are no odd nodes in the network, the network is traversable and the minimum distance is the sum of the arc distances.  Otherwise there will be an even number of odd nodes and the route inspection algorithm requires that we identify them and link them together in pairs in the most economic way.  The links selected will be repeated and the adding in of these extra arcs makes all the nodes even and the network traversable.

For more information on these procedures and the calculations behind them please see the previous posts on the route inspection problems.

## Linear Programming – Graphical Methods

Graphical Solution

When the L.P. problem involves only two decision variables its solution may be obtained by graphical means. Consider the earlier example. The shaded region OABCD contains all the points (x,y) which satisfy the constraints. The points form the set of feasible solutions. Clearly there are an infinite number of such points. We require that one which gives a maximum value of z. From z = 8x + 5y with rearranging we get y = – 8/5x + z/5, for different values of z this equation represents a family of straight lines each with a gradient -8/5. Z will be greatest when the family of parallel lines leave the feasible region i.e. at C (2.5,4.5). Point C’s values can be found using simultaneous equations of the lines which cross at that point. 3x +y = 12 and x + y = 7. By subtraction 2x = 5 \ x = 2.5 y = 4.5.

Hence Zmax = 8(2.5) + 5(4.5) = 42.5.

Integer Solutions

In the above problem we are producing bicycles and trucks. Yet our graphical solution gave us x being 2.5 and y being 4.5. Not very sensible, it is difficult to make half a truck! Some problems obviously must be integer solutions. In these cases we must “test” integer points within the feasible region. The integer point will usually be near the location were the family of profit lines left the feasible region. In our case near point C. Integer points near C yet still within the feasible region are (2,4), (2,5), (3,3). Putting these points in our objective function z = 8x + 5y gives z to be 36, 41 and 39 respectively. Hence our integer solution is a point (2,5) there the profit is £41.

If you find these difficult to follow there are lots more examples online.  Many of the best lessons used to come from the Open University Math’s models which were broadcast on the BBC.  I’m not sure if you can still find them as they are quite old now, but it might be worth checking the BBC’s archive – BBC iPlayer.   If you’re not able to access the site because of location restrictions then this site can help to access via a BBC iPlayer DNS service.

The Simplex Method

The Simplex Method is an algebraic rather than graphical approach to solving Liner Programming problems. The advantage of the Simplex Method is that it can cope with grater than 2 variables which cannot be solved graphically.

The first step is to state the general Linear Programming problem in a standard form.  We require a L.P. problem to be put into a form in which

1. the object function is to be maximised e.g. Maximise z =
2. all the variables are to be non-negative i.e. zero or positive because you cannot have negative amounts

III. all the constraints are to be equations (except for the non-negativity constraints) i.e. no >, , <,  but only =.

1. the right hand side constant of the constraints is positive or zero e.g. not x = 2 but x = 2 i.e. the RHS is +ve.

For further help see the below resources below:

Additional: BBC iPlayer Access – Free Trial Offer

## Real Life Maths – Bin Packing

Bin Packing

We have to fit a number of boxes of the same width and depth but different height into a rack. The rack is the same depth, divided into slots of the same width and of a fixed height.  There are 11 boxes, A to K, with heights as below

A – 8, B – 7, C – 4, D – 9, E – 6, F – 9, G – 5, H – 5, I – 6, J – 7, K – 9

The rack is 15 units high and we stack the boxes one on top of each other using as few slots as possible.

Similar problems might be cutting lengths of wood from standard lengths, or fitting vehicles into lanes on ferries. In each case we are trying to make the best possible use of the space available and avoid waste ( in the form of unused space above, off-cuts of wood and unfilled lanes).

These problems can be solved by trial and error, but it is worth looking for a more systematic approach (an algorithm).

For the Bin Packing problem there is no known algorithm that will always produce an optimal solution. There are many algorithms that find a good solution – known as heuristic algorithms.

Three are

I. Full-bin algorithm

Look for combinations of boxes to fill bins. Pack these boxes. For the rest place the next box to the next available space.

II. First-fit algorithm

Taking the boxes in the order listed, place the next box to be packed in the first available space.

III. First-fit decreasing algorithm

Reorder the boxes in decreasing size. Then apply the first-fit algorithm to this reordered list.

A method you could use for the first-fit or first-fit decreasing algorithm is:

Define a list of numbers, P, for the heights of the packages (ordered if necessary)

For the bin-packing example, P = {8,7,4,9,6,9,5,5,6,7,8}

Define a second set of numbers, B, for the space remaining in the bins. At the very worst this list will need to be as long as the list of packages.

For example B = {15,15,15,15,……}

Step 1. Take the first entry in P.

Step 2. Is it less than or equal to the first entry in B? Yes – Step 4

No – Step 3

Step 3. Go to the next B entry. Is it less than or Yes – Step 4

equal to this entry in B? No – Step 3

Step 4. Reduce the B entry by this amount.

Step 5 Any more entries in P? Yes – Take the next entry go to Step 2

No – Stop

The result in applying this algorithm is

P = {8, 7, 4, 9, 6, 9, 5, 5, 6, 7, 8}

B = {15,15,15,15,15,15,……}

7, 11, 9, 10, 9, 7

0, 2, 0, 5, 2

You can see how many bins have been used and how much free space there is in each and what packages are in each bin.

For more real world maths problems, you’ll find many online.  One of our favorite resources is the BBC Bitesize site, which at the time of writing has lots of these sorts of problems and other maths resources.   Most of them should be available internationally however much of the website is geo-restricted i.e you’ll not be able to access directly with US residential IPs. There is a solution to this though by hiding your real location using a proxy or VPN you can bypass these sorts of blocks – there’s an example in this post about downloading from the BBC abroad.

## Algorithm Primer

An algorithm is simply a sequence of precise instructions to solve a problem.  Precise means that there should be no ambiguity to any instruction or which instruction is next.

An example of an algorithm is Zeller’s Algorithm which is used to work out which day of the week a date is

Zeller’s Algorithm                                        Example      15/05/1991

Let Day number = D                                                        D = 15

Month number = M                                                         M = 5

and Year = Y                                                                  Y = 1991

If M is 1 or 2 add 12 to M and subtract 1 from Y.

Let C be the first 2 digits of Y                                          C = 19

and Y’ be the last 2 digits of Y                                        Y’ = 91

Add together the integer parts of

(2.6M5.39),(Y’4) and (C4), then                             7+22+4+15

add on D and Y’ and subtract 2C                                    +9138=101

{Note: Integer part of 2.3 = 2 and 6.7 = 6

1.7= 2 and 3.1 = 4}

Find the remainder when this quantity is divided by 7        1017=14r3

If the remainder is 0 – Sunday, 1 – Monday etc.

Communicating an Algorithm

We can communicate algorithms by ordinary language as above, by pseudo-code (stylised English), computer languages or flow charts.

Let’s take the example of finding the real roots of the quadratic equation

Ax2 + Bx + C = 0  (assume A0)

Using the equation

Using pseudo-code we could write

calc ; note its value;

if this is negative    [no real roots – stop]

else                       [calc root of value

calc (B+root) (2A)

calc (Broot) (2A) – stop].

For programmable calculators we could write

Casio Graphic                                                       Texas TI-81

?A                                                                     :Input A

?B                                                                     :Input B

?C                                                                     :Input C

D                                                    : D

(B+D) (2A)                                                     : (B+D) (2A)

:Pause

(BD) (2A)                                                     (BD) (2A)

{Note: These assume real roots}

In BBC Basic we could write

10      Input “A,B,C”;A,B,C

20      D=B*B4*A*C

30      IF D<0 THEN PRINT “No real roots”: STOP

40      PRINT (B+SQR(D))/(2*A)

50      PRINT (BSQR(D))/(2*A)

Using a flow chart we diagrammatically represent the sequence of steps in an algorithm.

Three types of boxes are common

1. The start stop box
2. The instruction box
3. The question box

Russian Peasant’s Algorithm

Example

Write down side by side the two numbers to be multiplied           163    24

Repeatedly [beneath the LHS number, write down the number    81      48

which is half the number above, ignoring remainders, beneath     40      96

the RHS number write down the number doubled] until you        20     192

reach 1 on the LHS.                                                                  10     384

5       768

2     1536

1     3072

Delete those rows where the number in the LHS is a multiple

of 2.  Add up the numbers left on the RHS                                163      24

81      48

40      96

20     192

10     384

5       768

2     1536

1     3072

This number is the answer                                                                3912

There’s a great demonstration of the Russian Peasants algorithm on the BBC learning websites, and if you’re lucky a couple of explanations on the BBC iPlayer archive.  For many maths lessons for children and adult learners it’s a great resource.  If you’re having trouble accessing from outside the UK then this BBC iPlayer DNS option works great and allows people to access all UK media sites without restrictions.

Another alternative is to use a resource which supplies residential IP addresses, although these aren’t specifically designed for streaming video they work pretty well.  There’s an example of them in this post ATC proxies explained be aware that they can be quite expensive though,

## Infinity is bigger than you think – Numberphile

Video Transcript

JAMES GRIME: We’re going to break a rule. We’re break one of the rules of Numberphile. We’re talking about something that isn’t a number. We’re going to talk about infinity. So infinity. Now like I said, infinity is not a number. It’s a idea. It’s a concept. It’s the idea of being endless, of going on forever. I think everyone’s familiar with the idea of infinity, even kids. You start counting 1, 2, 3, 4, 5– you might be five years old, but already you’re thinking, what’s the biggest number I can think of. And you go, oooh, it’s 20. You get a bit older, and you go, maybe it’s a million. It never ends, does it? ‘Cause you can keep adding 1. So that’s the idea of infinity. The numbers go on forever.

But I’m going to tell you one of the more surprising facts about infinity. There are different kinds of infinity. Some infinities are bigger than others. Let’s have a look. The first type of infinity is called countable. And I don’t like the name countable. And Brady gave me a little bit of a hmm, just then. Because if you’re talking about infinity, you can’t count infinity, can you? Because it goes on forever. I think it’s a terrible name. I prefer to call it listable. Can we list these numbers? All right.

Let’s do these simple numbers, 1, 2, 3–

BRADY HARAN: You’re not gonna do all of them, are you James?

JAMES GRIME: 4. How long have we got?

JAMES GRIME: Right. 5, 6– so you can list the whole numbers. So this is called countable. Listable, I prefer. What about the integers? All the integers. That’s all the negative numbers as well. So there’s 0. Let’s have that. But there’s 1 and minus 1, there’s 2 and minus 2, there’s 3, and minus 3. Now, that is an infinity as well. And in some sense, it’s twice as big, because there seems to be twice as many numbers. But it is infinity as well. They’re both infinity, and they’re both the same type of infinity. They both can be listed.

Perhaps more surprisingly, the fractions can be listed as well. But you have to be a bit clever about this. Let’s try and list the fractions. I’m going to write out a rectangle. 1 divided by 1. That’s a fraction. [INAUDIBLE]. Let’s have 1 divided by 2, 1/3, 1/4, 1/7– OK, that goes on. Let’s do the next row and have two at the top. 2/1, 2/2, 2/3, 2/4. Let’s do the next one. 3/1, 3/2. 4/6, 4/7. That goes on and we can keep going. So here, I’ve made some sort of an infinite rectangle array of fractions. Now if I want to make it a list like this, though, If I went row by row, you’re going to have a problem. If you go row by row, I’ll go– there’s 1, 1/2, 1/3, 1/5, 1/6, 1/7– and I’ll keep going forever. And I’m never going to reach the second row. I can’t list them. Not that way. You can’t list them that way. You’ll never reach the second row.

This is how you list them. Slightly more clever than that. You take the diagonal lines. Now, I can guarantee that every fraction will appear on one of those diagonal lines. And you list them diagonal by diagonal. So that’s the first diagonal. Then you list the second diagonal– there it is. Then you list the third diagonal, then you take the fourth diagonal, and the fifth. So eventually, you are going to do this every fraction. Every faction appears on a diagonal, and you’re going to list them. Now, if you take all the numbers, right? That’s the whole number line.

Let’s try that. Look, I’m going to draw it. It’s a continuous line of numbers. These are all your decimals. You’ve got 0 there in the middle, and you’ll go 1 and 2 and 3. But it has a 1/3. It will contain pi, and e, and all the irrational numbers as well. Can you list them? How do you list them? 0 to start with, and then 1? But hang on. We’ve missed a half. So we put in the half. Hang on, we’ve missed the quarter. We put in the quarter. But we’ve missed 0.237– so how do you list the real numbers? It turns out you can’t. In fact, rather remarkably, I can show you that we can’t list them, even though were talking about something so complicated as infinity.

JAMES GRIME: We need paper.

BRADY HARAN: We need an infinite amount of paper here, I think.

JAMES GRIME: (LAUGHING) It’s a big topic. Imagine we could list all the decimals, right? We can’t, actually. But pretend we can. What sort of– what would it look like? We’ll start with all the 0-point decimals. Let’s pick some decimals. 0.121– dot dot dot dot dot. Let’s pick the next one. Let’s say the next one is 0.221–. Next one, let’s do 0.31111129–. And let’s take another one, here. 0.00176–. Now I’m going to make a number. This is the number I’m going to make. I’m going to take the diagonals here.

I’m going to take this number and this number and this number and this number and this number. And I am going to write that down. So what’s that number I’ve made? It’s 0.12101– something, something, something. Now this is my rule. I’m going to make a whole new number from that one. This is the number I’m going to make. If it has a 1, I’m going to change it to a 2. And if it has a 2 or anything else, I will change it to a 1. So let’s try that. So I’m going to turn this into– 0-point. So if it has a 1, I’m going to turn it into a 2. If it’s anything else, I’m going to turn it into a 1. So that will be a 1. I’m going to change 1 here into a 2. I’m going to change that one into a 1. I’m going to change that one into a 2– that was my rule.

And I’ll make something new. That does not appear on the list. That number is completely different from anything else on the list, because it’s not the first number, because it’s different in the first place. It’s not the second number, because it’s different in second place. It’s not the third number, because it’s different in the third place. It’s not the fourth number because it’s different in the fourth place. It’s not the fifth number, because it’s different in the fifth place. You’ve made a number that’s not on that list. And so you can’t list all the decimals, in which case it is uncountable. It is unlistable. And that means it’s a whole new type of infinity. A bigger type of infinity.

BRADY HARAN: Surely we could, James, because all we’ve got to do is keep doing your game and making them and adding them to the list. And if we keep doing that, won’t we get there eventually?

JAMES GRIME: But you could then create another number that won’t be on that list. And so the guy who came up with is a German mathematician called Cantor. Cantor lived ’round about the turn of the 20th century. He was ridiculed for this. For this idea that there were different types of infinity, he was called a charlatan. And he was called– it was nonsense, it was called. And poor old Cantor was treated really badly by his contemporaries, and he spent a lot of his later life in and out of mental institutions, where he died, in the end. Near the end of his life, it was recognized. It was true. It was recognized. And he had all the recognition that he deserved.

BRADY HARAN: And now he’s on Numberphile.

JAMES GRIME: And now he’s on Numberphile, the greatest accolade of all. George Cantor.

For Watching Maths and technology Programs from places like the BBC – http://www.cedps.net/bbc-start-blocking-vpn-access-from-abroad/

## Math Antics – What Are Percentages?

Video Transcript

Hi! Welcome to Math Antics. Now that you know all about fractions, from watching all of our fractions videos, it’s time to learn about something called “percentages”. Percentages are super important. Have you ever been in a math class and heard another student ask the teacher: Um.. excuse me… teacher… Ah… when are we ever gonna use this stuff? Ya know… like in real life? Well when it come6s to percentages, the answer is one-hundred percent of the time. Well alright… maybe not a hundred percent of the time… but a lot!

Percentages are used every day to calculate things like: …how much sales tax you pay when you buy something. …how much something costs when it’s on sale. …how much fiber is in your granola bar. …or how much money you can make if you invest it in the stock market. That’s all real life stuff for sure. So, you can see that it’s really important to understand percentages and how we use them in math. Alright then… are you ready to learn the key to understanding percentages, or percents as they’re called for short? Drum roll please… A percent is a fraction!

Whaaaat? That’s right… a percent IS a fraction!

And since you already know all about fractions, learning about percents is gonna be easy. But a percent isn’t just any old fraction. A percent is a special fraction that always has 100 as the bottom number. If it’s a percent, then no matter what the top number is, the bottom number will be 100. In fact, because the bottom number of a percent is always 100, we don’t even write it. Instead, we use this handy little symbol (%) called a percent sign. Whenever you see this symbol after a number, it means the number is a percent. It’s really a fraction with 100 on the bottom, but it’s just being written in this more compact form. …like this number 15 here. It’s got the percent sign after it, so we read it as “15 percent”, and because a percent is really a fraction that always has 100 as the bottom number, we know that it means the same thing as 15 over 100. Percents make even more sense if you know what the word percent means. The prefix of the word (per) means “for each” or “for every”. Ya know like if someone said, “only one cookie per person”. And the root word (cent) is Latin for 100.

That’s why there’s 100 cents in a dollar. So, percent literally means “per 100” and that’s why they’re shortcuts for writing fractions that have 100 as the bottom number. Alright then, so whenever you see a percent like this, you know it can be replaced with (or converted) to a fraction. Let’s look at a few examples so you see the pattern. 3% means 3 over 100 10% means 10 over 100 25% means 25 over 100 and 75% means 75 over 100 These are percents… and these are the fractions that they stand for. There’s a few other interesting percents that we should take a look at. …like this one: 0% …can you have 0% ? Yes! 0% would just mean 0 over 100. It’s what we like to call a “zero fraction” cuz its value is just zero.

Remember, it’s okay to have zero on the top of a fraction, but not the bottom! Alright then, what about 100%. Well 100% just means 100 over 100. That’s what we like to call a “whole fraction”. The top number is the same as the bottom, so its value is just one whole, or 1. Okay then, 0% is just zero, and 100% is just 1. But what about numbers bigger than 100? Can you have 126% ? Yep, it works exactly the same way. 126% just means 126 over 100. And you know from the fractions videos, that’s what we call an “improper fraction”. The top number is bigger than the bottom number, so the fraction’s value will be greater than 1. Alright team, I want you to go out there and give me a-hundred and TEN percent effort in today’s game! But coach… it would be “improper” for us to give a-hundred and ten percent effort in today’s game.

Okay, so now you know the key to percentages. …that they’re just special fractions that always have 100 as the bottom number. But there’s one more thing that I need to tell you about in this video, and that’s decimals. Do you remember in the video about fractions and decimals that you can convert any fraction into its decimal value? Sometimes it was kind of tricky converting to a decimal if we had to divide the top number by the bottom number. But other times, like when we had “base-10” fractions, it was easy because decimal number places are made for counting base-10 fractions, (like tenths, hundredths and thousandths). Well guess what… Percents ARE base-10 fractions! They are hundredths because their bottom number is always 100. That means it’s really easy to re-write a percentage as a decimal number. You can do it the same way as we did in the base-10 fractions video.

For example, we know that 15% is just 15 over 100, right? That’s its fraction form. But it also has the decimal form 0.15 because THIS is the hundredths place and 0.15 means 15 hundredths. So, we can re-write 15% as a fraction (15 over 100) OR as a decimal (0.15) And now that you know WHY we can easily convert a percentage to a decimal, let me show you a really simple trick for doing it. First, you start with the number in percent form like this: 35% Next, you imagine where the decimal point should be in the number 35. It’s not shown, but if it was, it would be right here next to the ones place. (Now remember, 35 and 35.0 are the same value.) Now that you know where the decimal point is, just move it two number places to the left (away from the percent symbol) and draw it in right there. Last of all, once you have moved the decimal point, you erase the percent sign because you don’t have a percent anymore. Moving the decimal point two places to the left converted it into the decimal value of that percent. Let’s try converting a few more percents into their decimal values so you can get the hang of it. For 62 percent, we move the decimal point two places to the left and get 0.62 (Remember, we can put an extra zero in front of the decimal point to be a place holder and to make the decimal point easier to notice.) For 75 percent, we move the decimal point and get 0.75 For 99 percent, we move the decimal point to get 0.99

Pretty Cool, huh?

Okay, but what about 4% ? You might wonder how we can move the decimal point two places over when our number only has one digit. But all we need to do is use a zero as a place holder in the number place that’s missing. Then, when we move the decimal point two places over, we end up with the decimal value of 0.04. Now that makes sense because 4 is in the hundredths place and 4% is 4 over 100. And in the same way, 1% would just be 0.01. Again, we need that extra zero placeholder. Here’s a few more interesting examples: 0% would be just 0.00 And if we have 100% and we move the decimal point two places to the left, we end up with 1.00 But 1.00 is the same value as 1. That’s why 100% represent one whole. And if we have 142%, we move the decimal point to get 1.42 That’s a value greater than one which is what we’d expect because 142% is really an improper fraction (142 over 100) Its value should be greater than 1. Alright, so now you know that a percent is a special fraction that always has 100 as the bottom number. And you know that you can re-write percents in either their fraction form OR their decimal form. 25% is 25 over 100 or 0.25 But keep in mind that you could go the other way too. If someone gives you a fraction with 100 as the bottom number, you can re-write it in percent form. If you get 12 over 100, you can say that’s 12% And if you get 80 over 100, you can say that’s 80% OR… If you get the decimal 0.10, you can say that’s 10% and if you get the decimal 0.38, you can say that’s 38% So, that’s the key to percentages. They’re another way to write fractions and decimals. But there’s a lot more to learn about how they‘re used in math, and we’ll learn more about that in the next few videos.

But for now, you should be sure that you really understand the basics of percentages by doing the exercises for this section. Thanks for watching Math Antics, and I’ll see ya next time!