I have a simple problem, but because this "programming language" I am using is 32-bit and only supports basic functions such as addition, subtraction, multiplication, division, and concatenation (literally that's it), I am having some trouble.
For the input, I have a 16 digit number like so: 3334,5678,9523,4567
I want to then subtract 2 other random 16 digit numbers from this number and check if the first and last digits are 1.
For example, if the two other numbers are 1111,1111,1111,1111 and 1234,5678,9123,4565.
My final number would be: 0988,8888,9288,8891.
Here, the last number is 1, but the first number is 0, so the test would fail.
The issue is with 32-bit systems, there are massive errors due to not enough precision provided by the bits. What are some ways to bypass this issue?
If you're using a language like C or Java you should be able to use a long to create a 64 bit integer. If that's not possible you could divide the numbers into two 32 bit numbers, one to hold the upper half and one to hold the lower half.
Something like this:
//Each half is 8 digits to represent 8 of the 16
//Because of this each half should be less than 100000000
int upperHalf = 33345678;
int lowerHalf = 95234567;
//randomInt represents a function to generate a random
//integer equal to or greater than 0 and less than the
//argument passed to it
int randUpperHalf = randomInt(100000000);
int randLowerHalf = randomInt(100000000);
int lowerHalf = lowerHalf - randLowerHalf;
//If lowerHalf was a negative number you need to borrow from the upperHalf
if (lowerHalf < 0) {
upperHalf = upperHalf - 1;
lowerHalf = lowerHalf + 100000000;
}
upperHalf = upperHalf - randUpperHalf;
//Check that the first and last digits are 1
if ((upperHalf / 100000000) == 1 && (lowerHalf % 10) == 1) {
//The first and last digits are 1
}
Edit: Comments have been added to explain the code better. (lowerHalf % 2) == 1 has been changed to (lowerHalf % 10) == 1 and should now be able to tell if the number ends in a 1.
I have recently sat a computing exam in university in which we were never taught beforehand about the modulus function or any other check for odd/even function and we have no access to external documentation except our previous lecture notes. Is it possible to do this without these and how?
Bitwise AND (&)
Extract the last bit of the number using the bitwise AND operator. If the last bit is 1, then it's odd, else it's even. This is the simplest and most efficient way of testing it. Examples in some languages:
C / C++ / C#
bool is_even(int value) {
return (value & 1) == 0;
}
Java
public static boolean is_even(int value) {
return (value & 1) == 0;
}
Python
def is_even(value):
return (value & 1) == 0
I assume this is only for integer numbers as the concept of odd/even eludes me for floating point values.
For these integer numbers, the check of the Least Significant Bit (LSB) as proposed by Rotem is the most straightforward method, but there are many other ways to accomplish that.
For example, you could use the integer division operation as a test. This is one of the most basic operation which is implemented in virtually every platform. The result of an integer division is always another integer. For example:
>> x = int64( 13 ) ;
>> x / 2
ans =
7
Here I cast the value 13 as a int64 to make sure MATLAB treats the number as an integer instead of double data type.
Also here the result is actually rounded towards infinity to the next integral value. This is MATLAB specific implementation, other platform might round down but it does not matter for us as the only behavior we look for is the rounding, whichever way it goes. The rounding allow us to define the following behavior:
If a number is even: Dividing it by 2 will produce an exact result, such that if we multiply this result by 2, we obtain the original number.
If a number is odd: Dividing it by 2 will result in a rounded result, such that multiplying it by 2 will yield a different number than the original input.
Now you have the logic worked out, the code is pretty straightforward:
%% sample input
x = int64(42) ;
y = int64(43) ;
%% define the checking function
% uses only multiplication and division operator, no high level function
is_even = #(x) int64(x) == (int64(x)/2)*2 ;
And obvisouly, this will yield:
>> is_even(x)
ans =
1
>> is_even(y)
ans =
0
I found out from a fellow student how to solve this simplistically with maths instead of functions.
Using (-1)^n :
If n is odd then the outcome is -1
If n is even then the outcome is 1
This is some pretty out-of-the-box thinking, but it would be the only way to solve this without previous knowledge of complex functions including mod.
I'm looking at the documentation for random():
https://developer.apple.com/legacy/library/documentation/Darwin/Reference/ManPages/man3/srandomdev.3.html#//apple_ref/c/func/random
It returns successive pseudo-random numbers in the range from 0 to (2**31)-1.
I don't want it to return 0 ever.
I'm thinking about writing:
long rand = random() + 1;
But if I'm not mistaken, long can be 32-bits on a 32-bit processor. I guess I would risk stack overflow then.
What is the best approach to getting a random number between 1 and (2**31)-1?
NSUInteger r = arc4random_uniform(N) + 1;
This will generate a number between 1 and N. arc4random_uniform(N) generates a number between 0 and N-1.
You should have no problem with overflow.
long rand = 0;
while (rand == 0) {
rand = random();
}
This will almost absolutely certainly run exactly once. In a very, very rare case (that will never happen), it will run twice.
(Note that this is just a simplified version of how arc4random_uniform works. If you can use that function, as suggested by Jeff, you should.)
The maximum value returned by random() is RAND_MAX, so you can do this:
long rand = 1 + (random() % RAND_MAX);
When random() returns a value between zero and RAND_MAX-1, inclusive, you offset it by adding 1. When random() returns exactly RAND_MAX, modulo operator % converts the result to zero, so rand would be 1 again.
The drawback of this approach is that the probability of getting 1 becomes roughly twice as high as that of getting any other number.
I am using arc4random to generate a random number. I would like to generate a number between 1-9. How can I exclude 0?
int r = arc4random() % (9);
NSLog(#"Random Number: %d",r);
int r = (arc4random() % 8) + 1
You can use arc4random_uniform(), as in
arc4random_uniform(9) + 1
Generally, to generate a number between lo and hi (inclusive), you use:
arc4random_uniform(hi - lo + 1) + lo
If you don't want to use arc4random_uniform() and want to stick with arc4random(), noting that the resulting value from using modulus formula is not uniformly distributed, use
(arc4random() % (hi - lo + 1)) + lo
int r = arc4random() % 8 + 1;
See other answers (e.g., one from me) for why you probably don't want to use % for this task, and what you should use instead.
You could simply try repeatedly until you get a number in the range you want, throwing out numbers you don't want. This has the fancy name "acceptance-rejection method" in math. It's simple, and it works.
In case you're worried that this could take a long time, in theory it could. But this approach is used all the time in statistical applications. The probability of going through a while-loop N times decreases rapidly as N increases, so the average number of times through the loop is small.
I know the modulus (%) operator calculates the remainder of a division. How can I identify a situation where I would need to use the modulus operator?
I know I can use the modulus operator to see whether a number is even or odd and prime or composite, but that's about it. I don't often think in terms of remainders. I'm sure the modulus operator is useful, and I would like to learn to take advantage of it.
I just have problems identifying where the modulus operator is applicable. In various programming situations, it is difficult for me to see a problem and realize "Hey! The remainder of division would work here!".
Imagine that you have an elapsed time in seconds and you want to convert this to hours, minutes, and seconds:
h = s / 3600;
m = (s / 60) % 60;
s = s % 60;
0 % 3 = 0;
1 % 3 = 1;
2 % 3 = 2;
3 % 3 = 0;
Did you see what it did? At the last step it went back to zero. This could be used in situations like:
To check if N is divisible by M (for example, odd or even)
or
N is a multiple of M.
To put a cap of a particular value. In this case 3.
To get the last M digits of a number -> N % (10^M).
I use it for progress bars and the like that mark progress through a big loop. The progress is only reported every nth time through the loop, or when count%n == 0.
I've used it when restricting a number to a certain multiple:
temp = x - (x % 10); //Restrict x to being a multiple of 10
Wrapping values (like a clock).
Provide finite fields to symmetric key algorithms.
Bitwise operations.
And so on.
One use case I saw recently was when you need to reverse a number. So that 123456 becomes 654321 for example.
int number = 123456;
int reversed = 0;
while ( number > 0 ) {
# The modulus here retrieves the last digit in the specified number
# In the first iteration of this loop it's going to be 6, then 5, ...
# We are multiplying reversed by 10 first, to move the number one decimal place to the left.
# For example, if we are at the second iteration of this loop,
# reversed gonna be 6, so 6 * 10 + 12345 % 10 => 60 + 5
reversed = reversed * 10 + number % 10;
number = number / 10;
}
Example. You have message of X bytes, but in your protocol maximum size is Y and Y < X. Try to write small app that splits message into packets and you will run into mod :)
There are many instances where it is useful.
If you need to restrict a number to be within a certain range you can use mod. For example, to generate a random number between 0 and 99 you might say:
num = MyRandFunction() % 100;
Any time you have division and want to express the remainder other than in decimal, the mod operator is appropriate. Things that come to mind are generally when you want to do something human-readable with the remainder. Listing how many items you could put into buckets and saying "5 left over" is good.
Also, if you're ever in a situation where you may be accruing rounding errors, modulo division is good. If you're dividing by 3 quite often, for example, you don't want to be passing .33333 around as the remainder. Passing the remainder and divisor (i.e. the fraction) is appropriate.
As #jweyrich says, wrapping values. I've found mod very handy when I have a finite list and I want to iterate over it in a loop - like a fixed list of colors for some UI elements, like chart series, where I want all the series to be different, to the extent possible, but when I've run out of colors, just to start over at the beginning. This can also be used with, say, patterns, so that the second time red comes around, it's dashed; the third time, dotted, etc. - but mod is just used to get red, green, blue, red, green, blue, forever.
Calculation of prime numbers
The modulo can be useful to convert and split total minutes to "hours and minutes":
hours = minutes / 60
minutes_left = minutes % 60
In the hours bit we need to strip the decimal portion and that will depend on the language you are using.
We can then rearrange the output accordingly.
Converting linear data structure to matrix structure:
where a is index of linear data, and b is number of items per row:
row = a/b
column = a mod b
Note above is simplified logic: a must be offset -1 before dividing & the result must be normalized +1.
Example: (3 rows of 4)
1 2 3 4
5 6 7 8
9 10 11 12
(7 - 1)/4 + 1 = 2
7 is in row 2
(7 - 1) mod 4 + 1 = 3
7 is in column 3
Another common use of modulus: hashing a number by place. Suppose you wanted to store year & month in a six digit number 195810. month = 195810 mod 100 all digits 3rd from right are divisible by 100 so the remainder is the 2 rightmost digits in this case the month is 10. To extract the year 195810 / 100 yields 1958.
Modulus is also very useful if for some crazy reason you need to do integer division and get a decimal out, and you can't convert the integer into a number that supports decimal division, or if you need to return a fraction instead of a decimal.
I'll be using % as the modulus operator
For example
2/4 = 0
where doing this
2/4 = 0 and 2 % 4 = 2
So you can be really crazy and let's say that you want to allow the user to input a numerator and a divisor, and then show them the result as a whole number, and then a fractional number.
whole Number = numerator/divisor
fractionNumerator = numerator % divisor
fractionDenominator = divisor
Another case where modulus division is useful is if you are increasing or decreasing a number and you want to contain the number to a certain range of number, but when you get to the top or bottom you don't want to just stop. You want to loop up to the bottom or top of the list respectively.
Imagine a function where you are looping through an array.
Function increase Or Decrease(variable As Integer) As Void
n = (n + variable) % (listString.maxIndex + 1)
Print listString[n]
End Function
The reason that it is n = (n + variable) % (listString.maxIndex + 1) is to allow for the max index to be accounted.
Those are just a few of the things that I have had to use modulus for in my programming of not just desktop applications, but in robotics and simulation environments.
Computing the greatest common divisor
Determining if a number is a palindrome
Determining if a number consists of only ...
Determining how many ... a number consists of...
My favorite use is for iteration.
Say you have a counter you are incrementing and want to then grab from a known list a corresponding items, but you only have n items to choose from and you want to repeat a cycle.
var indexFromB = (counter-1)%n+1;
Results (counter=indexFromB) given n=3:
`1=1`
`2=2`
`3=3`
`4=1`
`5=2`
`6=3`
...
Best use of modulus operator I have seen so for is to check if the Array we have is a rotated version of original array.
A = [1,2,3,4,5,6]
B = [5,6,1,2,3,4]
Now how to check if B is rotated version of A ?
Step 1: If A's length is not same as B's length then for sure its not a rotated version.
Step 2: Check the index of first element of A in B. Here first element of A is 1. And its index in B is 2(assuming your programming language has zero based index).
lets store that index in variable "Key"
Step 3: Now how to check that if B is rotated version of A how ??
This is where modulus function rocks :
for (int i = 0; i< A.length; i++)
{
// here modulus function would check the proper order. Key here is 2 which we recieved from Step 2
int j = [Key+i]%A.length;
if (A[i] != B[j])
{
return false;
}
}
return true;
It's an easy way to tell if a number is even or odd. Just do # mod 2, if it is 0 it is even, 1 it is odd.
Often, in a loop, you want to do something every k'th iteration, where k is 0 < k < n, assuming 0 is the start index and n is the length of the loop.
So, you'd do something like:
int k = 5;
int n = 50;
for(int i = 0;i < n;++i)
{
if(i % k == 0) // true at 0, 5, 10, 15..
{
// do something
}
}
Or, you want to keep something whitin a certain bound. Remember, when you take an arbitrary number mod something, it must produce a value between 0 and that number - 1.