Learning how products are formed

We can now visualise multiplication of all kinds of numbers. But there is still something to be visualised – how can the product of 9 x 9 be expressed? It is simply 9 added 9 times. How do we get 81 as the product? This needs some exploration, and we will use number blocks and a few examples to understand this.

Example 1:  Using number blocks 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 is:

What is the number in decimal system that is 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9? What is Nine 9s (i.e., 9 sets of 9 units each) in the decimal number system?
Number in decimal system are made of units (  ), tens ( ), hundreds (   ) and so on. And there can only be a maximum of 9 units, 9 tens, 9 hundreds, 9 thousands, etc. in numbers.
Thus, there can’t be Nine 9s in a number – that is too many units in a number (9 sets of 9 units)! We need to make as many tens as possible from Nine 9s such that we don’t have more than 9 units in the product of 9 x 9.
Let us visually try to do this. It will be shown in a few steps for clarity:

Step 1: Let’s add the first two 9 units (9 + 9).

Step 2: Let’s add the second 9 (now 8 units) and the third 9 units.

Step 3: Let’s add the third 9 units (now 7 units) and the fourth 9 units.

Step 4: Let’s add the fourth 9 units (now 6 units) and the fifth 9 units.
We will skip the details and show the first and final numbers.

Step 5: Let’s add the fifth 9 units (now 5 units) and the sixth 9 units.

Step 6: Let’s add the sixth 9 units (now 4 units) and the seventh 9 units.

Step 7: Let’s add the seventh 9 units (now 3 units) and the eighth 9 units.

Step 8: Let’s add the eighth 9 units (now 2 units) and the ninth 9 units.

We have completed 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9, or 8 x 9. And the sum, or product, is 81:

Importantly, the process involved in getting the product of 9 x 9 is simple but very lengthy! It cannot be the way, and we need a better way.
Of course, this example was chosen to showcase how numbers are formed.
The following examples are simpler.

Example 2: 10 x 9
Using number blocks 10 x 9 is

This is a perfect number representation – 90 – it satisfies the two key conditions of a decimal number –

1. Only units, tens, hundreds, etc. is allowed as part of numbers.
2. The maximum number of units, tens, etc. in a number can only be 9.

In 90, there are 9 tens, so product of 10 x 9 is already a valid number.
Example 3: 8 x 21

Step 1:

Step 2:

Step 3:

Step 4:

Example 4: 11 x 11

Step 1:

Step 2:

Step 3:

Step 4:

Example 5: 12 x 14

Step 1:

Step 2:

Step 3:

Step 4:

Summary
Numbers are always expressed in strict rules of the decimal number system. Carry is also needed in finding products.

Excerpted from the book ‘Foundations of Multiplication (Mathematics as a language)’ by Sandeep Srivastava and Saloni Srivastava