How do you represent (decimal) integer 50 in binary?
How many bits must be "flipped" in order to capitalize a lowercase 'a' that is represented in ASC11?
How do you represent the (decimal) integer 50 in, oh, "hexadecimal," otherwise known as base-16? Recall that decimal is simply base-10, and binary is simply base-2. Infer from those base systems how to represent this one?
Please answer these questions for me.HELP.
To help you some:
Binary is only made up of 1's and 0's.This may help you understand binary conversion
Decimal is 0-9
Hexadecimal is 0-9, then A-F (so A would represent 10, B would be 11, etc up to F which is 15)
Converting from decimal to another base
Here some tips for you regarding conversion to binary:
What is 50 mod 2? What about 25 mod 2 and then 12 mod 2? What are your results if you continue this?
What does any number mod 2 (always) return as result? - 1 or 0
Do you realise any patterns? - You get the reversed binary number as result
Test case 50:
50 mod 2 = 0 - 6th digit
25 mod 2 = 1 - 5th digit
12 mod 2 = 0 - 4th digit
6 mod 2 = 0 - 3rd digit
3 mod 2 = 1 - 2nd digit
1 mod 2 = 1 - 1st digit
The remainders of the divisions concatenated and reverses are: 110010, which is 50 in binary.
Can this be also transformed to further bases? - Yes, as we see with trying to convert 50 to hexadecimal:
50 mod 16 = 2 - 2nd digit
3 mod 16 = 3 - 1st digit
The remainders again concatenated and reversed are 32, which conveniently is 50 in hexadecimal.
In general we can say to convert a number to an arbitrary base you must take the remainder of the number and the base and then divide the number by the base and do the same thing again. In a program this would look something like:
while the number is greater 0 do:
result = (number mod base) + result;
number = number div base;
Converting from any base to decimal
How do you convert a number from an arbitrary base into base 10? First let us do a test case with binary. Lets take the 50 from the previous example: 110010
The method to convert from binary is multiplying every digit with the base to the power of the position of it in the number and adding up the result. The enumeration of the positions begins with 0 at the least significant digit. Our previous number would then look something like this:
1 *2^5 + 1 *2^4 + 0 *2^3 + 0 *2^2 + 1 *2^1 + 0 *2^0
What simplifies to:
32 + 16 + 2 = 50
It also works with any other base, like our 32 from the previous example:
3 *16^1 + 2*16^0 = 48 + 2 = 50
In program this would look something like this:
from end of number to beginning do:
result = result + digit * (base ^ position)
Learning how to convert numbers from integers to binary.
I'm working on a fraction of .36 the binary for it is .01011... I understand that to get the binary if a fraction you times the number by 2 and read from the top number down.
So
.36 = 0 First number
.36 x 2 = .72 =1 , it's still below zero
.72 x 2 = 1.44 = 0, as it as it's above zero
1.44 x2 = 2.88 = 1, this is were I get thrown, is it becouse the .88 is closer to 1?
2.88 x2 = 5.76 =1
Giving me the .01011
So is it everything above .5 =1? so
I'm starting to play with floating point numbers so really need to know how to convert binary fractions
Your method is correct.
Some intuition: to convert an integer to base 2, you repeatedly take mod 2, giving your next digit, then divide by 2. Fractions are similar: think of it as converting to base 1/2: repeatedly take mod 1/2 (1 if the fractional part has 1/2, 0 otherwise), then divide by 1/2.
I have a matrix (size: 28 columns and 47 rows) with numbers. This matrix has an extra row that is contains headers for the columns ("ordinal" and "nominal").
I want to use the Gower distance function on this matrix. Here says that:
The final dissimilarity between the ith and jth units is obtained as a weighted sum of dissimilarities for each variable:
d(i,j) = sum_k(delta_ijk * d_ijk ) / sum_k( delta_ijk )
In particular, d_ijk represents the distance between the ith and jth unit computed considering the kth variable. It depends on the nature of the variable:
factor or character columns are
considered as categorical nominal
variables and d_ijk = 0 if
x_ik =x_jk, 1 otherwise;
ordered columns are considered as
categorical ordinal variables and
the values are substituted with the
corresponding position index, r_ik in
the factor levels. These position
indexes (that are different from the
output of the R function rank) are
transformed in the following manner
z_ik = (r_ik - 1)/(max(r_ik) - 1)
These new values, z_ik, are treated as observations of an
interval scaled variable.
As far as the weight delta_ijk is concerned:
delta_ijk = 0 if x_ik = NA or x_jk =
NA;
delta_ijk = 1 in all the other cases.
I know that there is a gower.dist function, but I must do it that way.
So, for "d_ijk", "delta_ijk" and "z_ik", I tried to make functions, as I didn't find a better way.
I started with "delta_ijk" and i tried this:
Delta=function(i,j){for (i in 1:28){for (j in 1:47){
+{if (MyHeader[i,j]=="nominal")
+ result=0
+{else if (MyHeader[i,j]=="ordinal") result=1}}}}
+;result}
But I got error. So I got stuck and I can't do the rest.
P.S. Excuse me if I make mistakes, but English is not a language I very often.
Why do you want to reinvent the wheel billyt? There are several functions/packages in R that will compute this for you, including daisy() in package cluster which comes with R.
First things first though, get those "data type" headers out of your data. If this truly is a matrix then character information in this header row will make the whole matrix a character matrix. If it is a data frame, then all columns will likely be factors. What you want to do is code the type of data in each column (component of your data frame) as 'factor' or 'ordered'.
df <- data.frame(A = c("ordinal",1:3), B = c("nominal","A","B","A"),
C = c("nominal",1,2,1))
Which gives this --- note that all are stored as factors because of the extra info.
> head(df)
A B C
1 ordinal nominal nominal
2 1 A 1
3 2 B 2
4 3 A 1
> str(df)
'data.frame': 4 obs. of 3 variables:
$ A: Factor w/ 4 levels "1","2","3","ordinal": 4 1 2 3
$ B: Factor w/ 3 levels "A","B","nominal": 3 1 2 1
$ C: Factor w/ 3 levels "1","2","nominal": 3 1 2 1
If we get rid of the first row and recode into the correct types, we can compute Gower's coefficient easily.
> headers <- df[1,]
> df <- df[-1,]
> DF <- transform(df, A = ordered(A), B = factor(B), C = factor(C))
> ## We've previously shown you how to do this (above line) for lots of columns!
> str(DF)
'data.frame': 3 obs. of 3 variables:
$ A: Ord.factor w/ 3 levels "1"<"2"<"3": 1 2 3
$ B: Factor w/ 2 levels "A","B": 1 2 1
$ C: Factor w/ 2 levels "1","2": 1 2 1
> require(cluster)
> daisy(DF)
Dissimilarities :
2 3
3 0.8333333
4 0.3333333 0.8333333
Metric : mixed ; Types = O, N, N
Number of objects : 3
Which gives the same as gower.dist() for this data (although in a slightly different format (as.matrix(daisy(DF))) would be equivalent):
> gower.dist(DF)
[,1] [,2] [,3]
[1,] 0.0000000 0.8333333 0.3333333
[2,] 0.8333333 0.0000000 0.8333333
[3,] 0.3333333 0.8333333 0.0000000
You say you can't do it this way? Can you explain why not? As you seem to be going to some degree of effort to do something that other people have coded up for you already. This isn't homework, is it?
I'm not sure what your logic is doing, but you are putting too many "{" in there for your own good. I generally use the {} pairs to surround the consequent-clause:
Delta=function(i,j){for (i in 1:28) {for (j in 1:47){
if (MyHeader[i,j]=="nominal") {
result=0
# the "{" in the next line before else was sabotaging your efforts
} else if (MyHeader[i,j]=="ordinal") { result=1} }
result}
}
Thanks Gavin and DWin for your help. I managed to solve the problem and find the right distance matrix. I used daisy() after I recoded the class of the data and it worked.
P.S. The solution that you suggested at my other topic for changing the class of the columns:
DF$nominal <- as.factor(DF$nominal)
DF$ordinal <- as.ordered(DF$ordinal)
didn't work. It changed only the first nominal and ordinal column.
Thanks again for your help.
I don't really understand how modulus division works.
I was calculating 27 % 16 and wound up with 11 and I don't understand why.
I can't seem to find an explanation in layman's terms online.
Can someone elaborate on a very high level as to what's going on here?
Most explanations miss one important step, let's fill the gap using another example.
Given the following:
Dividend: 16
Divisor: 6
The modulus function looks like this:
16 % 6 = 4
Let's determine why this is.
First, perform integer division, which is similar to normal division, except any fractional number (a.k.a. remainder) is discarded:
16 / 6 = 2
Then, multiply the result of the above division (2) with our divisor (6):
2 * 6 = 12
Finally, subtract the result of the above multiplication (12) from our dividend (16):
16 - 12 = 4
The result of this subtraction, 4, the remainder, is the same result of our modulus above!
The result of a modulo division is the remainder of an integer division of the given numbers.
That means:
27 / 16 = 1, remainder 11
=> 27 mod 16 = 11
Other examples:
30 / 3 = 10, remainder 0
=> 30 mod 3 = 0
35 / 3 = 11, remainder 2
=> 35 mod 3 = 2
The simple formula for calculating modulus is :-
[Dividend-{(Dividend/Divisor)*Divisor}]
So, 27 % 16 :-
27- {(27/16)*16}
27-{1*16}
Answer= 11
Note:
All calculations are with integers. In case of a decimal quotient, the part after the decimal is to be ignored/truncated.
eg: 27/16= 1.6875 is to be taken as just 1 in the above mentioned formula. 0.6875 is ignored.
Compilers of computer languages treat an integer with decimal part the same way (by truncating after the decimal) as well
Maybe the example with an clock could help you understand the modulo.
A familiar use of modular arithmetic is its use in the 12-hour clock, in which the day is divided into two 12 hour periods.
Lets say we have currently this time: 15:00
But you could also say it is 3 pm
This is exactly what modulo does:
15 / 12 = 1, remainder 3
You find this example better explained on wikipedia: Wikipedia Modulo Article
The modulus operator takes a division statement and returns whatever is left over from that calculation, the "remaining" data, so to speak, such as 13 / 5 = 2. Which means, there is 3 left over, or remaining from that calculation. Why? because 2 * 5 = 10. Thus, 13 - 10 = 3.
The modulus operator does all that calculation for you, 13 % 5 = 3.
modulus division is simply this : divide two numbers and return the remainder only
27 / 16 = 1 with 11 left over, therefore 27 % 16 = 11
ditto 43 / 16 = 2 with 11 left over so 43 % 16 = 11 too
Very simple: a % b is defined as the remainder of the division of a by b.
See the wikipedia article for more examples.
I would like to add one more thing:
it's easy to calculate modulo when dividend is greater/larger than divisor
dividend = 5
divisor = 3
5 % 3 = 2
3)5(1
3
-----
2
but what if divisor is smaller than dividend
dividend = 3
divisor = 5
3 % 5 = 3 ?? how
This is because, since 5 cannot divide 3 directly, modulo will be what dividend is
I hope these simple steps will help:
20 % 3 = 2
20 / 3 = 6; do not include the .6667 – just ignore it
3 * 6 = 18
20 - 18 = 2, which is the remainder of the modulo
Easier when your number after the decimal (0.xxx) is short. Then all you need to do is multiply that number with the number after the division.
Ex: 32 % 12 = 8
You do 32/12=2.666666667
Then you throw the 2 away, and focus on the 0.666666667
0.666666667*12=8 <-- That's your answer.
(again, only easy when the number after the decimal is short)
27 % 16 = 11
You can interpret it this way:
16 goes 1 time into 27 before passing it.
16 * 2 = 32.
So you could say that 16 goes one time in 27 with a remainder of 11.
In fact,
16 + 11 = 27
An other exemple:
20 % 3 = 2
Well 3 goes 6 times into 20 before passing it.
3 * 6 = 18
To add-up to 20 we need 2 so the remainder of the modulus expression is 2.
The only important thing to understand is that modulus (denoted here by % like in C) is defined through the Euclidean division.
For any two (d, q) integers the following is always true:
d = ( d / q ) * q + ( d % q )
As you can see the value of d%q depends on the value of d/q. Generally for positive integers d/q is truncated toward zero, for instance 5/2 gives 2, hence:
5 = (5/2)*2 + (5%2) => 5 = 2*2 + (5%2) => 5%2 = 1
However for negative integers the situation is less clear and depends on the language and/or the standard. For instance -5/2 can return -2 (truncated toward zero as before) but can also returns -3 (with another language).
In the first case:
-5 = (-5/2)*2 + (-5%2) => -5 = -2*2 + (-5%2) => -5%2 = -1
but in the second one:
-5 = (-5/2)*2 + (-5%2) => -5 = -3*2 + (-5%2) => -5%2 = +1
As said before, just remember the invariant, which is the Euclidean division.
Further details:
What is the behavior of integer division?
Division and Modulus for Computer Scientists
Modulus division gives you the remainder of a division, rather than the quotient.
It's simple, Modulus operator(%) returns remainder after integer division. Let's take the example of your question. How 27 % 16 = 11? When you simply divide 27 by 16 i.e (27/16) then you get remainder as 11, and that is why your answer is 11.
Lets say you have 17 mod 6.
what total of 6 will get you the closest to 17, it will be 12 because if you go over 12 you will have 18 which is more that the question of 17 mod 6. You will then take 12 and minus from 17 which will give you your answer, in this case 5.
17 mod 6=5
Modulus division is pretty simple. It uses the remainder instead of the quotient.
1.0833... <-- Quotient
__
12|13
12
1 <-- Remainder
1.00 <-- Remainder can be used to find decimal values
.96
.040
.036
.0040 <-- remainder of 4 starts repeating here, so the quotient is 1.083333...
13/12 = 1R1, ergo 13%12 = 1.
It helps to think of modulus as a "cycle".
In other words, for the expression n % 12, the result will always be < 12.
That means the sequence for the set 0..100 for n % 12 is:
{0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,0,[...],4}
In that light, the modulus, as well as its uses, becomes much clearer.
Write out a table starting with 0.
{0,1,2,3,4}
Continue the table in rows.
{0,1,2,3,4}
{5,6,7,8,9}
{10,11,12,13,14}
Everything in column one is a multiple of 5. Everything in column 2 is a
multiple of 5 with 1 as a remainder. Now the abstract part: You can write
that (1) as 1/5 or as a decimal expansion. The modulus operator returns only
the column, or in another way of thinking, it returns the remainder on long
division. You are dealing in modulo(5). Different modulus, different table.
Think of a Hash Table.
When we divide two integers we will have an equation that looks like the following:
A/B​​ =Q remainder R
A is the dividend; B is the divisor; Q is the quotient and R is the remainder
Sometimes, we are only interested in what the remainder is when we divide A by B.
For these cases there is an operator called the modulo operator (abbreviated as mod).
Examples
16/5= 3 Remainder 1 i.e 16 Mod 5 is 1.
0/5= 0 Remainder 0 i.e 0 Mod 5 is 0.
-14/5= 3 Remainder 1 i.e. -14 Mod 5 is 1.
See Khan Academy Article for more information.
In Computer science, Hash table uses Mod operator to store the element where A will be the values after hashing, B will be the table size and R is the number of slots or key where element is inserted.
See How does a hash table works for more information
This was the best approach for me for understanding modulus operator. I will just explain to you through examples.
16 % 3
When you division these two number, remainder is the result. This is the way how i do it.
16 % 3 = 3 + 3 = 6; 6 + 3 = 9; 9 + 3 = 12; 12 + 3 = 15
So what is left to 16 is 1
16 % 3 = 1
Here is one more example: 16 % 7 = 7 + 7 = 14 what is left to 16? Is 2 16 % 7 = 2
One more: 24 % 6 = 6 + 6 = 12; 12 + 6 = 18; 18 + 6 = 24. So remainder is zero, 24 % 6 = 0