Arithmetic - Basics 4

Thursday, May 04, 2006

Now that we have seen Addition, Subtraction and Division, let us also learn easier methods to multiply numbers.

Multiplication - Criss-cross method:
Example 1) Suppose you want to multiply 18 x 19.

Step 1) Multiply the right hand digits i.e 8 x 9 = 72, write down the 2 and carry 7
Step 2) Cross multiply the digits as shown in figure and sum them, and add the carry, i.e. (9 x 1) + (8 x 1) + 7 = 24, write down 4 and carry 2
Step 3) Multiply the left hand digits, i.e 1 x 1 and add it with carry i.e 2, hence 1 + 2 = 3, write it down.

Example 2) Multiply 1234 x 45

Step 1) Multiply the right hand digits, 4 x 5 = 20, write 0 and carry 2
Step 2) Multiply the next digits, 3 x 5 and 4 x 4 as shown in figure, add the results, add the carry, result 33, write down 3 and carry 3.
Step 3) Multiply the next digits, 5 x 2 and 3 x 4 as shown in figure, add the results i.e 10 + 12, add the carry, result 25, write down 5 and carry 2.
Step 4) Multiply the next digits, 5 x 1 and 4 x 2 as shown in figure, add the result, add the carry, result 15, write down 5 and carry 1.
Step 5) Multiply the left hand digits, 4 x 1 = 4, add carry, result 5, write it down, so final answer is 55530.

Squares:
When we square a number we multiply the number with itself, hence we can use the above method. But there is a pattern when we square a number with criss-cross method, let us look at it.

Example 1) Suppose you want to square 24, i.e 24 x 24, so from the criss-cross method.
= 2 x 2 / 4 x 2 + 4 x 2 /4 x 4
= 2^2 / 2 (4 x 2) / 4^2
= 4 / 16 / 16
= 4 + 1 / 6 + 1 / 6
= 576

So we observe it is a^2 / 2ab / b^2, don't mistake, there is no addition sign.

Example 2) Square of 29 = ?
Now a = 2 and b = 9, by the above observation we substitute the values.
= 2^2 / 2 (2 x 9) / 9^2
= 4 / 36 / 81 ( hear 8 is a carry, so add it to the preceding number )
= 4 / 36 + 8 / 1 ( after adding 8 to 36 you get 44, carry 4 to the preceding number)
= 4 / 44 / 1
= 4 + 4 / 4 / 1
= 8 / 4 / 1
= 841

This method is pretty fast if you get a hang of it. Basically square up to 30 are to be remembered, this method will come handy to find out square of numbers greater than 30.

Cubes:
Two steps to find out the cube of a number.

Step 1) We have to write down 4 numbers, first we have to take the cube of the 10's digit of the given number, for other 3 numbers we have to take the ratio of the units digit and 10's digit of the given number and write them in geometrical progression.
Step 2) Now double the 2nd and 3rd numbers and write them below the numbers respectively, then add the first row to the second.

This is little bit tricky, nothing to do with criss-cross method.

Example 1): 13^3
Step 1) The 10's digit is 1, cube of 1 is 1, hence write down 1, now take the ratio of 3 and 1, i.e 3:1, so the next numbers should be 3/1 of the previous once, write down the other 3 numbers in g.p, i.e 1 3 9 27
Step 2) Now double the 2nd and the 3rd numbers i.e 3 and 9 and write the results below them, respectively, i.e 6 and 18. Now add the two rows. Final answer is 2197

Example 2): 29^3
Step 1) The 10's digit is 2, cube of 2 is 8, hence write down 8, now take the ratio of 9:2, i.e the other 3 numbers will be 9/2 of the previous number, write down the other 3 numbers in g.p, i.e 8 x 9/2 = 36, 36 x 9/2 = 162, 162 x 9/2 = 729.
Step 2) Now double the 2nd and the 3rd numbers i.e 36 and 162 and write the results below them, respectively, i.e 72 and 324. Now add the two rows. Final answer 24389

Hope everything is clear. If you find it difficult to grasp, go through again, still if you don't get, forget it, as cubes of higher numbers are rarely needed.