Introduction:
1. 1S3
2. Samik Sen (for the first term, Richard Timoney for 2nd and 3rd)
3. Course description
4. Binary numbers
5. Symbolic computing
Course description
Binary numbers
- Can write a number in many ways; roman numerals for example.
- In general we use the decimal system, since we have 10 fingers?
- When we write 7892 we mean:
  
  7*10³+8*10²+9*10+2.
  
  Similarly when we write 13 we mean:
  
  1*10+3.
  
  Since we are dealing with powers of 10 we say we are working with base 10. If we were in base 2, which is binary, could write the same number as:
  
  1*2³+1*2²+0*2¹+1*2⁰,
  
  or 1101. Note that 10⁰ = 2⁰ = 1.
- Why are we interested in doing this? Computers. Here we deal with 0's and 1's. We have 2 digits, or effectively 2 fingers. So base 2 makes more sense. It is the same thing though, as we will see.
- We want to be able to convert from our normal base 10 numbers to base 2 and back again. For the moment we'll just do an example. If we want to write 101 in base 2, we find the highest power of 2 which is lower than this. The powers are 1,2,4,8,16,32,64,128,.... So we see that 2⁶ = 64 is going to be 1. We can then add 32 to this and still be less than 101. If we do this we have 101-64-32=5 left. So 16 and 8 must have 0 coefficients since if they were 1 we would over shoot our target. 5 is, in powers of 2, 4+1. This means that 101 is
  
  1*2⁶+1*2⁵+1*2²+1*2⁰ = 1100101.
  
  To do a simpler example to see what's going on more, we look at 7. What is this in terms or powers of 2? Again we have 1,2,4,8,16,32,.... Since 8 is greater then 7, we can only use 1,2 and 4. This means that
  
  7 = 1*2²+1*2¹+1*2⁰ = 111.
  
  We can do this in a systematic manner. Given a number, 75 say, we divide it by two, with perhaps a remainer. Here this leads to 37 with remainder 1. We then divide 37, which leads to 18, with remainer 1. We repeat leading to 9, with remainder 0, 4 with remainder 1, 2 with remainder 0 and 0 with remainder 1.
  Now list the remainders in reverse order:1001011, which equals:
  
  1*2⁶+0*2⁵+0*2⁴+1*2³+0*2²+1*2¹+1*2⁰ =
  
  1*64+0*32+0*16+1*8+0*4+1*2+1 = 64+8+2+1 = 75.
- We will also look at base 8 and base 16, but we'll come back to that later.
- In this way we can represent number of interest, but we are being slightly deceptive. A computer only has a finite amount of space. This means that we cannot capture p say, which has infinitely many digits, but at the same time having 20 digits is more than enough is most cases. That then is the problem. We can capture more than enough digits for nearly any calculation, but it is not the same as the actual thing. In many cases we use only 8 digits, since it most cases this is easily enough, but it does mean that if we say x=1000000000 and add one to it we get 1000000000. This can of course lead to problems. We'll come back to this later too.
Symbolic computation(or Souped up calculators):
- We can work things out computationally, we can represent numbers using bits and add them together, but we can also manipulate things algebraically. This means that we can deal with more abstract concepts than numbers in the form of bits.
- Integration, which we will look at later, is about finding the area under a curve. There are simple rules for doing this. For example is the curve is described by f(x) = x, we know that the area under it is given by [(x²)/2]. We could use computational power and break the area into many rectangles to work out the area under the curve using one of various numerical schemes OR we could symbolically know that x goes to [(x²)/2]. In this way we build in some effective intelligence.
- You can think of this as a souped up calculator. We first learn to add up and multiply. Having done so we can use calculators which automate the process. When multiplying 48*56, it's easier to do it on a calculator(though you should be able to work it out in your head..think of it as (50-2)*56, then you get 2800-112=2688). In the same way we can differentiate and integrate but why waste time doing this. You want to be thinking about the problem you are working about rather than spending time on pages of calculations where it's easy to make a mistake somewhere.
- You still need to know something about the maths since otherwise you won't know what the various calculator functions are. If you don't know your numbers then you won't know when you want to multiply them or add them.
Computational Learning:
- This is more a personal interest of mine. In the same way that calculations can be computerised so can various parts of lecturing. Why spend time taking down notes which are typed in anyway. So having them on the web seems to me to be more efficient. Ok take notes, but the basic content which i write down will be on the web. Some calculations won't be in as much detail, so compare what is on the web to what you take down for a couple of weeks to see how much you want to take down.
- A computerised sytem is more interactive. Since can ask give problems to which it knows the solution. Can include more questions to practice on.
- Can have common questions which people ask with their answers since people tend to not ask as many questions as they should. You can look at the notes in you own time and check up on things.
- Need to know what you find useful since it's not much use if you don't think it is. The idea is to create a simple system in a few courses, see how it works, then expand to more and more courses till have a nice system which is available to everyone for free. From the lecturer point of view it frees up time too!
- That said at the moment have nothing. Will have the notes up, but the questions and answers programs for this course are not done yet; see how tricky that proves out to be. For the moment there are some web pages:
```
www.sunybroome.edu/~antonakos_j/j.html
```

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On 9 Oct 2002, 14:25.