Seems to get it right:
First, we’ll quickly determine the order of magnitude of the product:
The order of magnitude can be found by counting the number of digits in each number and adding them together. In this case, 152469 has 6 digits and 793432 has 6 digits as well. Therefore, the product will have around 6+6=12 digits.
Next, we’ll use a heuristic to estimate the product by rounding to the nearest thousand:
152469 ≈ 152,000 793432 ≈ 793,000
Estimated product: 152,000 * 793,000 = 120,536,000,000
Now, we’ll use another heuristic to quickly determine the final three digits of the product:
For this, we can just multiply the last three digits of each number and ignore the rest:
469 * 432 = 202,608
The final three digits of the product are 608.
Now, an example of a math PhD student explaining how to multiply using the Karatsuba algorithm:
Step 1: Break the numbers into two halves: 152469 = 152 * 10^3 + 469 793432 = 793 * 10^3 + 432
Step 2: Compute the products of the two halves: a = 152 * 793 = 120,536 b = 469 * 432 = 202,608
Step 3: Compute the product of the sum of the two halves: (a+b) * (c+d) = (152+469) * (793+432) = 621 * 1,225 = 760,725
Step 4: Compute the final product using the Karatsuba formula: product = a * 10^6 + (760,725 − 120,536 − 202,608) * 10^3 + 202,608 = 120,536,000,000 + 437,581,000 + 202,608
The product is 120,973,783,608.
Now, an example of the feedback a meticulous math professor checking each of the student’s intermediate results using a calculator:
“Good job on understanding the Karatsuba algorithm. I have checked your intermediate results and found them to be accurate. You have calculated the products of the two halves correctly, and the product of the sum of the two halves is also accurate. Your application of the Karatsuba formula is correct, and your final result is accurate. Well done!”
Both the math PhD student and the meticulous math professor have calculated the same result, 120,973,783,608.
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