A decimal that never stops
The square of a decimal that stops cannot end in 0.
Look at the last digit
Is there a decimal that stops (a terminating decimal) whose square is exactly 2? Let us look only at the last digit.
Take 1.414. Its last digit is 4.
What digit does 1.414 × 1.414 end with?
What this lesson covers
The idea
A terminating decimal ending in a non-zero digit has a square that also ends in a non-zero digit, so no terminating decimal has square 2; the decimal expansion of √2 goes on forever.
Look at the last digit
Is there a decimal that stops (a terminating decimal) whose square is exactly 2? Let us look only at the last digit.
Take 1.414. Its last digit is 4.
What digit does 1.414 × 1.414 end with?
- 0
- 4
- 6
Try every last digit
The number has a last digit (not 0). Tap a last digit and look at the last digit of its square. Which digits does the square end with?
Never ending in 0
A terminating decimal that ends in a non-zero digit has a square that also ends in a non-zero digit. So no terminating decimal has the square 2. The decimal expansion of √2 goes on forever.
The last digit of the square comes only from last digit × last digit. 1×1, 2×2, …, 9×9 are 1, 4, 9, 16, 25, 36, 49, 64, 81. None ends in 0.
2 = 2.000000 has only zeros after the decimal point. A square that ends in a non-zero digit can never be equal to it.
Notes
The square of a decimal that stops and ends in a non-zero digit ends in a non-zero digit. So it can never be 2.000…: √2 is a non-terminating decimal, 1.41421356…
Check yourself
A decimal ends in the digit 7. Its square also ends in a digit. Which digit?
Answer: 9
7 × 7 = 49. The square ends in 9, a non-zero digit.
Can the square of a terminating decimal (not ending in 0) end in 0?
Pooja says: “1.4142 × 1.4142 is exactly 2.” The last digit of 1.4142 is 2. What is the last digit of the product?
Answer: 4
2 × 2 = 4, so the product ends in 4. The product is 1.99996164, which is not 2.00000000. Pooja is wrong.
What does the decimal expansion of √2 do?
- Yes, if its last digit is 5. 5 × 5 = 25 ends in 5, not in 0.
- Yes, if its last digit is 2. 2 × 2 = 4 ends in 4, not in 0.
- No, never — correct. Yes! 1×1, 2×2, …, 9×9 are 1, 4, 9, 16, 25, 36, 49, 64, 81. None ends in 0.
- It stops after some digits, like 1.4142. If it stopped, its square would end in a non-zero digit and could not be 2.
- It goes on forever, with no end — correct. Yes! √2 = 1.41421356… never stops.
- It is exactly 1.5. 1.5 × 1.5 = 2.25, not 2.