Introduction
When you use your computer or your calculator to make difficult calculations, or when you see your child being easily the following multiplication:
68
x 43
-----------
204
272
-----------
2924
We have to remember the suffering of the ancients throughout history to calculate the operations of arithmetic, however left us great works certifying that they were geniuses.
History tells us about two operations of multiplication, Ancient Egyptian multiplication and Russian peasants multiplication.
Ancient Egyptian multiplication
Here is how 43 is multiplied by 68
- Start with number 1 in the first column and keep doubling (the first column has powers of two), the largest power of two less than or equal to the the first number(43).
- Start with the second number (68) in the second column, keep doubling the number in the second column.
- Subtract the largest power of two less than or equal to the first number(43),
43 - 32 = 11,
subtract the largest power of two less than or equal to the remainder(9),
11 - 8 = 3
repeat,
3 - 2 = 1
repeat until nothing remained,
1 - 1 = 0
now you see that 43 = 32 + 8 + 2 + 1.
- To get result, Check the numbers in the second column corresponding to 32, 8, 2, 1 and add them.
Russian peasants multiplication
See how 43 is multiplied by 68
- Write each number at the head of a column.
- Divide the number in the first column by 2, flooring the quotient (drop the remainder), until there is nothing left to divide.
- Keep doubling the number in the second column, until you have doubled it as many times as you divided the number in the first column.
- To get result, add up all the numbers in the second column that are next to an odd number in the first column.
Test operations of multiplication
To test the previous operations of multiplication begin a new project with one form, put the following controls on the form:
- two controls of TextBox: txtFirst and txtSecond to enter two numbers.
- two controls of CheckedListBox for Ancient Egyptian multiplication: lstEgyptian1 and lstEgyptian2.
- two controls of CheckedListBox for Russian peasants multiplication: lstRussian1 and lstRussian2.
- two controls of Label to print the result: EgyptianResult and RussianResult.
- three controls of Buttons: btnEgyptianCalc, btnRussianCalc to get the result and btnExit to exit the program.
About the Code
Fill first list:
private void FillListEgyptian1()
{
int newVlue = 1;
numFirst =
Convert.ToInt32(txtFirst.Text);
lstEgyptian1.Items.Clear();
//add powers of two (less than numFirst) to 'lstEgyptian1'
while (newVlue <= numFirst)
{
lstEgyptian1.Items.Add(newVlue);
newVlue = 2 *
newVlue; //the first column has powers of two
}
}
Fill second list:
private void FillListEgyptian2()
{
numSecond = Convert.ToInt32(txtSecond.Text);
lstEgyptian2.Items.Clear();
lstEgyptian2.Items.Add(numSecond);
for(int i = 1; i <
lstEgyptian1.Items.Count; i++)
{
numSecond =
numSecond * 2; //Keep doubling the second number
lstEgyptian2.Items.Add(numSecond);
}
}
Check numbers at first column:
private void CheckEgyptianNumbers()
{
int newNumber = numFirst;
int i = lstEgyptian1.Items.Count;
do
{
i--;
int binNumber = (int)(lstEgyptian1.Items[i]);
if(binNumber <= newNumber) //is power of 2 < newNumber?
{
newNumber
= newNumber - binNumber;
lstEgyptian1.SetItemChecked(i,
true); //check it
lstEgyptian2.SetItemChecked(i,
true);
}
}
while (i > 0);
}
Add numbers to get the result:
private void GetEgyptianMultiplication()
{
long TheResult = 0;
for(int i = 0; i
< lstEgyptian2.Items.Count; i++)
{
if(lstEgyptian2.GetItemChecked(i) == true)
{
TheResult
= TheResult + (int)(lstEgyptian2.Items[i]);
EgyptianResult.Text
= TheResult.ToString();
}
}
}
You can read the code of Russian peasants multiplication.
We must not forget what previous generations have given us the knowledge, said Isaac Newton "I stand on the shoulders of my predecessors".