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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsWhole NumbersWhole Numbers: Multiplication and Division

Multiplication : Simplified Procedure for Large Numbers

This page extends the multiplication in first principles into a simplified procedure for multiplication of large numbers, which is called multiplication by place-value with regrouping.

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Consider multiplication of 12 and 3.

This is performed as, combine 12 repeatedly 3 times and count the combined quantity.

Considering the multiplication 12xx3.

Combining 12 repeatedly 3 times is shown in the figure. There are 3 tens and 6 units which together form value 36.

Considering the multiplication 12xx3. The number 12 is given in the place value format in the figure.

The repeated addition is visualized and the multiplication is performed as illustrated in the figure.

The units place is multiplied as 2xx3=6 and the tens place is multiplied as 1xx3 = 3. The result of the multiplication is 36.

Consider the multiplication 26xx2. That is, 26 is repeated twice as shown in the figure. Counting the two quantities together, there are 4 tens and 12 units. The combined value is 52.

Considering the multiplication 26xx2. That is, 26 is repeated twice and combined as shown in the figure. Note that 10 units are regrouped together to form the 1 ten. So the result of the multiplication is 52.

Considering the multiplication of 26xx2. The number 26 is given in the place value format in the figure. A simplified procedure Multiplication by Place-value with regrouping is given in the figure.

In the units place, 6 and 2 are multiplied to 12. The 10 unit is converted to 1 ten and moved to the tens position. This regrouping 10 of lower place-value to 1 of the immediate higher place-value is carry over. The remaining units 2 is retained.

In the tens place, 2 and 2 are multiplied to 4 and the carry over is added to that. So, the tens place of result is 4+1=5.

The result is 52.

Consider the multiplication: 34xx4. One 34 is shown in the figure.

To multiply, the quantity 34 can be repeated 4 times or the simplified procedure "multiplication by place-value with regrouping" can be used.

Considering the multiplication: 34xx4. The value 34 is repeated 4 times and combined as shown in the figure. The combined quantity makes 12 tens and 16 units.

The 10 tens are regrouped into 1 hundred. The 10 units are regrouped into 1 ten. Then the number of hundreds, tens, and units are counted to the result 136.

Considering the multiplication: 34xx4. The numbers are given in the place value in the figure.

A simplified procedure, Multiplication by Place-value is given in the figure.

The digit in the units place is multiplied by the multiplier. The tens formed is carried over to the tens position,
Then the digit in the tens place is multiplied by the multiplier. The hundreds formed is carried over to the hundreds position.

The result of the multiplication is 136.

Consider the multiplication of 47 and 23. The numbers are shown in the place value format in the figure.

The multiplication in place value is illustrated. The product is 1081.

Multiplication by Place-value with Regrouping - Simplified Procedure : Two numbers are multiplied as illustrated in the figure. Note: The procedure combines 10 units into 1 ten as carry over and so on for higher place-values.

Solved Exercise Problem:

What is 284 xx 20

• 568
• 5680
• 5680

The answer is "5680"

Solved Exercise Problem:

What is 11xx123?

• 253
• 1353
• 1353

The answer is "1353"

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