I was in a meeting last week when the presenters had a 32 by 9 matrix of items on the board. He was trying to calculate the number of total cells in this matrix. I blurted out 288. He looked at the board again, and after a few seconds, agreed that there was 288 items in the matrix. Later on, he was calculating the totoal of some small scale costs figures on the board, $9.25/hour for 12 hours. Again, I blurted out $111.00. He looked back at me with an expression that I have never seen before.
After the meeting, he asked me how I was able to multiply so quickly. I told him that I don’t multiply like other people. In back in school, I learned that they teach elementary multiplication inefficiently. As some of you may remember, here’s elementary schools teach you how to multiply:
Problem: 32 X 9
Step 1: Rearrange the numbers into a vertical format.
32
X 9
--------
Step 2: Multiply 2 X 9. The answer, 18, is more than 9, so the second digit is placed under the line, and carry the other digit. So, the 8 is placed in the answer and the 1 is carried.
1
32
X 9
--------
8
Step 3: Multiply 3 X 9, then add the carry. Again, the answer, 28 is more than 9. Like before, The 8 is placed in the next spot in the answer and the 2 is carried.
21
32
X 9
--------
88
Step 4: Now, there is no more digits in the first number, so the carry (2) drops down in the next place in the answer
21
|32
X | 9
--------
288
Now, you have your answer, 288. But, I definitely did not do this in my head, because it is a lot of work. What did I do? I took advantage of a multiplication property called distributivity. What I did was break down the problem into pieces and distribute the pieces over simple addition.
So, a multiplication of larger numbers can break down into one or more simpler representations. For example, 32 X 9 can be broken down into simple pieces:
32 X 9 = (30 + 2) X 9
= (30 X 9) + (2 X 9)
= 270 + 18
= 288
Now, for me, multiplying 30 X 9 is trivial, but for some, further (or even a different) breakdown is needed. So, now:
32 X 9 = (30 + 2) X 9
= (30 X 9) + (2 X 9)
= (3 X 10 X 9) + (2 X 9)
= (3 X 90) + (2 X 9)
= 288
Now, addition is not the only way. If it can work for you, subtraction is a good choice, too. Subtraction is just another form of addition. So, I can round the 9 up to 10, then compensate using subtraction:
32 X 9 = 32 X (10 - 1)
= (32 X 10) - (32 X 1)
= 320 - 32
= 288
In general, I like to break down problems into multples of 10 (10, 20, 30…) because it’s real trivial to multiply by 10. Add a zero at the end of your operand!
For more complex problems, the method is the same, but more addition is needed. For the money problem:
$9.25 X 12 = ($9.00 + $0.25) X (10 + 2)
= ($9.00 X 10) + ($0.25 X 10) + ($9.00 X 2) + ($0.25 X 2)
= $90.00 + $2.50 + $18.00 + $0.50
= $111.00
Or the subtraction method (somthing simpler):
$9.25 X 12 = ($10.00 X 12 ) - ($0.75 X 12)
= $120.00 - $9.00
= $111.00
So, with this powerful method of multiplication, you can impress your friends and peers, too!
Well this is pretty much what I do naturally as I suck at maths. Never learnt the name for what I was doing tho. Only problem I have is I forget what my previous calculation was before adding something else, so have to start again.
Well this is pretty much what I do naturally as I suck at maths. Never learnt the name for what I was doing tho. Only problem I have is I forget what my previous calculation was before adding something else, so have to start again.
Try
23.12 x
14.23
in your head
@Alex,
Ahh, two decimal numbers. The technique still works, but I would make it simpler.
To solve 23.12 x 14.23 in my head, I would multiply each operand by a multiple of 10, in this case, 100. This way, the problem becomes multiplication of two integers. However, when you are done, you need to put your decimal places back. In this example, I would need to divide by 10000 (100 x 100).
My father used to talk about how my grandmother would multiply four digit numbers in her head as the kids would work them out on paper. He said her responses where almost instantaneous. She had an 8th grade education. Is there a trick to multiplying four digit numbers or was she some kind of savant?
@John L
Your grandmother probably used one of many algorithms used for fast multiplication. For example, the algorithm that I described in the post is called Booth’s Algorithm. I did not know at the time I wrote the post, I did not know it had a name or even if it existed. I just used it naturally because it made sense. Your grandmother probably did the same.
that’s cool!!!
this is the coolest meathod i have ever seen!!!!!!!
if you are having trouble remembering the numbers from the previous calculations then i found a trick for you!!
i am very good @ seeing in 3d in my head so i make block numbers in my head of the numbers so that they become an image which is easier to remember
if you can’t do that well try this try to write the numbers of something in the room not literally but in you mind take for instance 7*182 = 1374 so 7*100=700 and pretend to write than on the ceiling then 7*80= 560 so pretend to write that on the floor
then 2*7=14 so add 14 to whats on the floor then add that sum to whats on the ceiling!!!
it does take a good pit of practice but when you can do it your friends will be in awe when you try to explain this to them.