What is a way to simulate division using only multiplication and whole numbers?

Question

What is a way to simulate division using only multiplication and whole numbers?

The issue at hand:

The stack-building API that I'm currently using in rapidweaver has limitations when it comes to division or floats, making it challenging to perform certain mathematical operations.

Insights into the API's rules:

This particular API is incorporated within html, css, and javascript files, which are compiled into actual html, css, and javascript during the project file generation process.

Every mathematical operation within this API is enclosed in:

%( math )%
For example, to calculate (1 + 3) * 6, the format would be:

%( %( 1 + 3 )% * 6 )%

Attempting to use %( 1 + 3 * 6 )% would lead to an error, as only one operation is allowed within each %()%.
Additionally, it's possible to use variables that represent user input

%id=variableName%
Therefore, if the goal is to double the user input, the code would look like:

%( %id=variableName% * 2 )%

The objective at hand:

%( %id=CB_seconds% / 2 )% OR %( %id=CB_seconds% * 0.5 )%

Unfortunately, the API restricts the use of the above expressions for reasons unknown...

The desired outcome is to take the user-defined input %id=CB_seconds%, and divide it by half. The user will input a value between 0 and 10,000 as an integer. Both the original value and the halved value need to be displayed separately. One workaround could involve asking the user to enter a value between 0 and 5,000 and then multiplying it by two. However, this solution is not ideal as it increases complexity for the user who is setting the animation length.

The Query:

Is there a method to obtain half of a number using only integers and multiplication/addition/subtraction, without prior knowledge of the number?

For more information on the Stacks API documentation, visit -

javascript html css algorithm math

Answer 1

Answer №1

It is impossible to create a function using only addition and multiplication.

This is a mathematical concept:

If you were to try to create a function using only addition and multiplication, it would eventually lead to a polynomial function such as:

a/b=f(a,b)

When you analyze this function as n grows to infinity

You will find that f(n,n)=n/n=1

However, polynomials always approach +infinity, -infinity, or 0, making it impossible for such a function to exist.

Answer 2

It is impossible to create a function using only addition and multiplication.

This is a mathematical concept:

If you were to try to create a function using only addition and multiplication, it would eventually lead to a polynomial function such as:

a/b=f(a,b)

When you analyze this function as n grows to infinity

You will find that f(n,n)=n/n=1

However, polynomials always approach +infinity, -infinity, or 0, making it impossible for such a function to exist.

Answer 3

Answer №2

Just like multiplication can be thought of as a series of additions, division can be seen as a series of subtractions. In order to achieve this, you must have the ability to loop and compare values. For dividing a by b, assuming both are integers and a is greater than b, the pseudocode would look something like this:

division = 1
c = a - b
while (c > b)
  c = c - b
  division = division + 1
end while

After running this code, you will have a / b = division and the remainder as c.

Let's take an example with a = 10 and b = 3:

division = 1
c = 10 - 3 = 7

c (= 7) > 3 --> Continue
  c = 7 - 3 = 4
  division = 2

c (= 4) > 3 --> Continue
  c = 4 - 3 = 1
  division = 3

c (= 1) < 3 --> Stop

After running this, you will find that division = 3 and c = 1, which means 10 / 3 = 3 (+ 1).

It seems the Stacks API allows for comparison. If it also supports looping, then you can use the aforementioned method.

In conclusion, it's quite silly that a division operation is not readily available.

Answer 4

Just like multiplication can be thought of as a series of additions, division can be seen as a series of subtractions. In order to achieve this, you must have the ability to loop and compare values. For dividing a by b, assuming both are integers and a is greater than b, the pseudocode would look something like this:

division = 1
c = a - b
while (c > b)
  c = c - b
  division = division + 1
end while

After running this code, you will have a / b = division and the remainder as c.

Let's take an example with a = 10 and b = 3:

division = 1
c = 10 - 3 = 7

c (= 7) > 3 --> Continue
  c = 7 - 3 = 4
  division = 2

c (= 4) > 3 --> Continue
  c = 4 - 3 = 1
  division = 3

c (= 1) < 3 --> Stop

After running this, you will find that division = 3 and c = 1, which means 10 / 3 = 3 (+ 1).

It seems the Stacks API allows for comparison. If it also supports looping, then you can use the aforementioned method.

In conclusion, it's quite silly that a division operation is not readily available.

Answer 5

Answer №3

Admittedly, this may not be the most efficient coding technique, but one approach could be to iterate through each number from 1 to 5000, doubling it, and then checking to see if it matches the original input number (also checking for the number + 1 to account for rounded fractions). With only 5000 operations needed, the impact on performance should be minimal.

Answer 6

Admittedly, this may not be the most efficient coding technique, but one approach could be to iterate through each number from 1 to 5000, doubling it, and then checking to see if it matches the original input number (also checking for the number + 1 to account for rounded fractions). With only 5000 operations needed, the impact on performance should be minimal.

Answer 7

Answer №4

Fresh response

Given the absence of loops, a binary search must be implemented. The approach involves:

CB_copy = CB_seconds;
CB_half = 0;
for ( exp = 13 ; exp > 0 ; exp- - ) {
    if ( CB_copy > 1 << exp ) {
        CB_half += 1 << (exp-1);
        CB_copy -= 1 << exp;
    }
}

However, lacking conditionals compels a revision using available comparisons:

CB_copy = CB_seconds;
CB_half = 0;
for ( exp = 13 ; exp > 0 ; exp- - ) {
    CB_half += ( 1 << (exp-1) ) * ( CB_copy > 1 << exp );
    CB_copy -= ( 1 << exp ) * ( CB_copy > 1 << exp );
}

Since loops are not supported, manual unwinding is essential, leading to 28 lines of code:

CB_copy = CB_seconds;
CB_half = 0;
CB_half += ( 1 << 12 ) * ( CB_copy > 1 << 13 );
CB_copy -= ( 1 << 13 ) * ( CB_copy > 1 << 13 );
CB_half += ( 1 << 11 ) * ( CB_copy > 1 << 12 );
CB_copy -= ( 1 << 12 ) * ( CB_copy > 1 << 12 );
. . .
CB_half += ( 1 << 1 ) * ( CB_copy > 1 << 2 );
CB_copy -= ( 1 << 2 ) * ( CB_copy > 1 << 2 );
CB_half += ( 1 << 0 ) * ( CB_copy > 1 << 1 );
CB_copy -= ( 1 << 1 ) * ( CB_copy > 1 << 1 );

In the absence of bit shifting, substitution with custom values from a plist is advised. Replace 1 << k with twoExp{k} for a set of k values in the plist:

CB_half += ( twoExp12 ) * ( CB_copy > twoExp13 );
CB_copy -= ( twoExp13 ) * ( CB_copy > twoExp13 );
etc

Lastly, convert the above into appropriate syntax:

%CB_copy = %id=CB_seconds% %
%CB_half = 0%
%CB_half = %( CB_half + %( twoExp12 * %( CB_copy > twoExp13 )% )% )% %
%CB_copy = %( CB_copy - %( twoExp13 * %( CB_copy > twoExp13 )% )% )% %
. . .
%CB_half = %( CB_half + %( twoExp0 * %( CB_copy > twoExp1 )% )% )% %
%CB_copy = %( CB_copy - %( twoExp1 * %( CB_copy > twoExp1 )% )% )% %

28 lines of code dedicated to halving a value. Best wishes.

Previous solution

Implementing GrimRepear1908's concept involves iterating up to 5000 and checking for the desired value through doubling. In a more conventional language, the logic appears as follows:

for repeatIndex in (0..5001):
  CB_half += repeatIndex*( repeatIndex*2==CB_seconds || repeatIndex*2==CB_seconds+1 )

For clarity within the API, keeping one operation per line is recommended, resulting in:

for repeatIndex in (0..5001):
  doubled = repeatIndex*2
  doubledLessOne = doubled-1
  CB_evenTarget = doubled==CB_seconds
  CB_oddTarget = doubledLessOne==CB_second
  isTarget = CB_evenTarget || CB_oddTarget // is 0 or 1
  multed = repeatIndex * isTarget
  CB_half = multed + CB_half

And in the specific syntax of the language, the code would be structured as follows:

%[repeat 5001]%
doubled = %( %id=repeatIndex% *2 )% %
doubledLessOne = %( %id=doubled% -1 )%
CB_evenTarget = %( %id=doubled% == %id=CB_seconds% )%
CB_oddTarget = %( %id=doubledLessOne% == %id=CB_second% )%
isTarget = %( %id=CB_evenTarget% || %id=CB_oddTarget% )%
multed = %( %id=repeatIndex% * %id=isTarget% )%
CB_half = %( %id=multed% + %id=CB_half% )%
%[endrepeat]%

Nuclearman's suggestion of a binary search method may offer brevity in the code, but considering the complexity involved in a simple loop, that approach is not currently explored.

Answer 8

Fresh response

Given the absence of loops, a binary search must be implemented. The approach involves:

CB_copy = CB_seconds;
CB_half = 0;
for ( exp = 13 ; exp > 0 ; exp- - ) {
    if ( CB_copy > 1 << exp ) {
        CB_half += 1 << (exp-1);
        CB_copy -= 1 << exp;
    }
}

However, lacking conditionals compels a revision using available comparisons:

CB_copy = CB_seconds;
CB_half = 0;
for ( exp = 13 ; exp > 0 ; exp- - ) {
    CB_half += ( 1 << (exp-1) ) * ( CB_copy > 1 << exp );
    CB_copy -= ( 1 << exp ) * ( CB_copy > 1 << exp );
}

Since loops are not supported, manual unwinding is essential, leading to 28 lines of code:

CB_copy = CB_seconds;
CB_half = 0;
CB_half += ( 1 << 12 ) * ( CB_copy > 1 << 13 );
CB_copy -= ( 1 << 13 ) * ( CB_copy > 1 << 13 );
CB_half += ( 1 << 11 ) * ( CB_copy > 1 << 12 );
CB_copy -= ( 1 << 12 ) * ( CB_copy > 1 << 12 );
. . .
CB_half += ( 1 << 1 ) * ( CB_copy > 1 << 2 );
CB_copy -= ( 1 << 2 ) * ( CB_copy > 1 << 2 );
CB_half += ( 1 << 0 ) * ( CB_copy > 1 << 1 );
CB_copy -= ( 1 << 1 ) * ( CB_copy > 1 << 1 );

In the absence of bit shifting, substitution with custom values from a plist is advised. Replace 1 << k with twoExp{k} for a set of k values in the plist:

CB_half += ( twoExp12 ) * ( CB_copy > twoExp13 );
CB_copy -= ( twoExp13 ) * ( CB_copy > twoExp13 );
etc

Lastly, convert the above into appropriate syntax:

%CB_copy = %id=CB_seconds% %
%CB_half = 0%
%CB_half = %( CB_half + %( twoExp12 * %( CB_copy > twoExp13 )% )% )% %
%CB_copy = %( CB_copy - %( twoExp13 * %( CB_copy > twoExp13 )% )% )% %
. . .
%CB_half = %( CB_half + %( twoExp0 * %( CB_copy > twoExp1 )% )% )% %
%CB_copy = %( CB_copy - %( twoExp1 * %( CB_copy > twoExp1 )% )% )% %

28 lines of code dedicated to halving a value. Best wishes.

Previous solution

Implementing GrimRepear1908's concept involves iterating up to 5000 and checking for the desired value through doubling. In a more conventional language, the logic appears as follows:

for repeatIndex in (0..5001):
  CB_half += repeatIndex*( repeatIndex*2==CB_seconds || repeatIndex*2==CB_seconds+1 )

For clarity within the API, keeping one operation per line is recommended, resulting in:

for repeatIndex in (0..5001):
  doubled = repeatIndex*2
  doubledLessOne = doubled-1
  CB_evenTarget = doubled==CB_seconds
  CB_oddTarget = doubledLessOne==CB_second
  isTarget = CB_evenTarget || CB_oddTarget // is 0 or 1
  multed = repeatIndex * isTarget
  CB_half = multed + CB_half

And in the specific syntax of the language, the code would be structured as follows:

%[repeat 5001]%
doubled = %( %id=repeatIndex% *2 )% %
doubledLessOne = %( %id=doubled% -1 )%
CB_evenTarget = %( %id=doubled% == %id=CB_seconds% )%
CB_oddTarget = %( %id=doubledLessOne% == %id=CB_second% )%
isTarget = %( %id=CB_evenTarget% || %id=CB_oddTarget% )%
multed = %( %id=repeatIndex% * %id=isTarget% )%
CB_half = %( %id=multed% + %id=CB_half% )%
%[endrepeat]%

Nuclearman's suggestion of a binary search method may offer brevity in the code, but considering the complexity involved in a simple loop, that approach is not currently explored.

What is a way to simulate division using only multiplication and whole numbers?

The issue at hand:

Insights into the API's rules:

The objective at hand:

The Query:

Answer №1

It is impossible to create a function using only addition and multiplication.

Answer №2

Answer №3

Answer №4

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