Reasoning vs Computation Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Without computing, determine whether 997×1003997 \times 1003 is greater than, less than, or equal to 100021000^2. Then verify by computation.

Solution

  1. 1
    Reasoning: Write 997=1000−3997 = 1000 - 3 and 1003=1000+31003 = 1000 + 3. This is the difference-of-squares pattern: (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.
  2. 2
    So 997×1003=10002−32=1,000,000−9=999,991<1,000,000997 \times 1003 = 1000^2 - 3^2 = 1{,}000{,}000 - 9 = 999{,}991 < 1{,}000{,}000.
  3. 3
    Computation confirms: 997×1003=999,991997 \times 1003 = 999{,}991.

Answer

997×1003=999,991<1,000,000=10002997 \times 1003 = 999{,}991 < 1{,}000{,}000 = 1000^2
Structural reasoning (recognising the difference-of-squares form) gives the answer and a general insight, while computation merely confirms the specific case. Reasoning is often faster and more illuminating than brute-force calculation.

About Reasoning vs Computation

Reasoning is the process of understanding why a mathematical fact is true and how ideas connect, while computation is the mechanical process of calculating an answer — both are essential but serve different purposes.

Learn more about Reasoning vs Computation →

More Reasoning vs Computation Examples

Example 2 medium

Use reasoning (not direct computation) to find the units digit of [formula].

Example 3 easy

Without long division, determine whether [formula] is greater or less than [formula]. Explain your r

Example 4 medium

By reasoning (not expanding), determine the degree of the polynomial [formula].

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