Square Root of 7 to 100 Decimal Places: Exact Digits and Step-by-Step Proof

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Before desktop chips executed floating-point instructions in nanoseconds, human computers used the manual square root long division method. Based on the identity (20p + x)x, this manual algorithm extracts digits one position at a time without trial-and-error drift.

Here is how to calculate the first three decimal places of √7 by hand:

Step 1: Group the digits into pairs.

Starting from the decimal point, group digits moving left and right: 07 . 00 00 00.

Step 2: Find the largest integer whose square is ≤ 7.

2² = 4 (since 3² = 9, which exceeds 7). The first root digit is 2.

Subtract 4 from 7 to get a remainder of 3. Bring down the first pair of zeros (00) to create the working dividend: 300.

Step 3: Calculate the next digit.

Double the current root (2 × 2 = 4). We search for a single digit x such that (40 + x) × x ≤ 300.

Testing x = 6: 46 × 6 = 276.

Testing x = 7: 47 × 7 = 329 (too large).

Our second root digit is 6. Subtract 276 from 300 to leave 24. Bring down the next zero pair: 2400.

Step 4: Extract the second decimal place.

Our current root is 26 (ignoring the decimal). Double it to get 52.

Find a digit y such that (520 + y) × y ≤ 2400.

Testing y = 4: 524 × 4 = 2096.

Testing y = 5: 525 × 5 = 2625 (too large).

The third root digit is 4. Subtract 2096 from 2400 to yield 304. Bring down two zeros: 30400.

Step 5: Extract the third decimal place.

Our running root is 264. Double it to get 528.

Find digit z such that (5280 + z) × z ≤ 30400.

Testing z = 5: 5285 × 5 = 26425.

Testing z = 6: 5286 × 6 = 31716 (too large).

The fourth root digit is 5. The manual extraction gives 2.645, matching the analytical target.