can anyone help me visualize multidigit multiplication ?
for example let us consider this question
245
X345
here when we multiply 4 at tens place with 5 with at ones place we do it as if it onesXones multiplication but place the product at tens place and same with 3 at hundreds place with 4 at tens place. can anyone anwer why this idea works and what's the logic? How to visualize it?
it works because of the distributive property of multiplication and addition.
345 = 300+40+5
so,
345*245 = 5*245 + 40*245 + 300*245
Sure! Visualizing multidigit multiplication can be helpful in understanding the logic behind it. When multiplying numbers like 245 and 345, we can break it down into three separate multiplications:
1. Ones' place multiplication: Multiply the ones digit of the second number (in this case, 5) by each digit of the first number (starting from the right) and write the products in the ones' place.
2. Tens' place multiplication: Multiply the tens digit of the second number (in this case, 4) by each digit of the first number (starting from the right) and write the products one place value to the left (tens' place) of the ones' place.
3. Hundreds' place multiplication: Multiply the hundreds digit of the second number (in this case, 3) by each digit of the first number (starting from the right) and write the products one place value to the left (hundreds' place) of the tens' place.
Finally, add up all the products to get the result.
Here's a step-by-step breakdown of how to visualize and solve the example:
245
X345
_________________
1225 (ones' place multiplication: 5 x 5 = 25, write 5 in the ones' place, carry over 2 to the tens' place)
000 (tens' place multiplication: 4 x 5 = 20, write 0 in the ones' place, carry over 2 to the hundreds' place)
1225 (tens' place multiplication continued: 4 x 4 = 16, add 2 carried to get 18, write 8 in the tens' place, carry over 1 to the hundreds' place)
+ 81675 (hundreds' place multiplication: 3 x 5 = 15, write 5 in the hundreds' place, carry over 1 to the thousands' place)
_________________
84625
So, the answer is 84,625.
The logic behind this method is based on the concept of place value. Each digit in a number has a specific place value (ones, tens, hundreds, etc.). When multiplying multidigit numbers, we need to consider the place value of each digit to correctly calculate the product. By breaking down the problem into smaller multiplications and placing the resulting products in the appropriate place values, we maintain the integrity of the place value system and accurately calculate the final result.