Multiplication

Division and multiplication are about groups and grouping. 

3 * 4 

if I have 3 groups and each group has 4 units, how many units do I have in total? 

Or if I have 4 groups and 3 units in each group, how many units do I have in total?

( Due to the commutative property in multiplication, both sentences are correct, but the commutative property is not the subject of this post ) 

In multiplication, the numbers being multiplied are called factors, and the result is called the product.

Long Multiplication Operation

Have you ever wondered what "carry" means in the classic multiplication method taught in schools?

43.5

When we perform this operation, it comes to the sum of 43 added 5 times.

The product is 20 tens and 15 ones, which equals 215.

When we multiply 5 by 3, it means we are adding three units five times, resulting in 15 units. We write the 5 units in the ones place of the product, and we carry over the remaining 10 units as 1 in the tens place. This carried-over 10 units is converted into one ten, which we will add to the other tens in the multiplication.

When we multiply 5 by 4, we are actually adding four tens five times. Five times four tens makes twenty tens. We also had one ten carried over, so in total, we end up with twenty-one tens.

Twenty-one tens means 210. I write 1 in the tens place for the 210's tens, and the remaining 200 goes in the hundreds place, with 2 written there, since there are no more hundreds left to multiply

5 times 3 units = 15

5 times 40 = 200

Total = 215

Example

24x36

The meaning of 36 multiplied by 24 is to find the sum of 36 instances of 24. I start with determining what 6 times 24 is. (Think of 24 as 4 units and 2 tens, which is 20.) How much is 6 times 4? 6 times 4 is 24. I write the 4 from the 24 in the units place and need to carry over the remaining 20, which is 2 tens, to add to the other tens. I make a note of the number to carry over on the side. You see a 2 on the side, but its actual meaning is 20

I multiply 6 by 2 (considering the place values, you are actually multiplying 6 by 20, so 6 * 20 = 120. I also had 2 tens carried over from the units, making a total of 140. The place after the units place is the tens place, so I will write the 40 of 140 as 4 in the tens place and carry over the remaining hundred again.

I don't have another hundred to multiply by 6, so I only have the carried-over 100. I write the carried-over hundred in the hundreds place.

Thus, we have found the total of 6 times 24

Now it's time to find the total of 30 times 24.

Apparently, you will multiply 3 by 24, but due to the place value, you are actually multiplying 30 by 24, so the real result is 10 times what you see. That's why you should either shift one place over to the tens place or add a 0 to the units place and continue with your multiplication. You should choose and continue with whichever method is easier for you.

In this example, let's choose the method of shifting places that is also used in my country, and multiply in that way. I multiplied 4 by 3, which is 4*3=12, and I wrote down the 2 and said there is one carry-over. What actually happened? Due to the place value, 3 represents 30, so the value of 4 times 30 is 120. I'm writing in the tens place, so I write the 20 of 120 as 2 in the tens place, and I say there is one carry-over for the remaining 100, which I will carry over to the other hundreds.

I multiply 3 by 2, but due to the place values, I am actually finding 30 times 20. 30 times 20 equals 600, and I also had 100 carried over from the tens, making a total of 700. If I write 7 in the hundreds place, it carries the meaning of 700.

* Represents multiplication

Why do we move over places in multiplication?

We move over digits in multiplication to align the place values correctly when multiplying multi-digit numbers. When multiplying numbers in the decimal system, each digit represents a different place value (units, tens, hundreds, etc.). By moving over the digits, we ensure that the products of the corresponding place values are correctly aligned and can be added together to get the final result

42 x 25

This multiplication means finding the sum of 42, twenty-five times.

The result here is not the product of 2 and 42, but actually the product of 20 and 42. The sum of 20 times 42 is 840. Instead of writing 840, we shift the digit and write 84, but there is actually a hidden zero there, and when we add that number to 210, by writing the 4 under the tens place, we give it the meaning of 40. We gave the 8 the meaning of 800.

Example

How can we multiply without moving over places?

You need to multiply the numbers by paying attention to their place values and then add them together.

Why does multiplying a number by zero result in zero?

5 x 0

if I have 5 groups and each group has 0 units, how many units do I have in total?  0 Units..

Or if I have 0 groups and 5 units in each group, how many units do I have in total? 0 Units..

( In reference to the second sentence, if I have no groups, then having 5, 10, 100, etc., units in each group doesn't make sense. )

I have zero of 100, meaning I have none.