Square Numbers Table

What Are Square Numbers?

A square number (also called a perfect square) is an integer that can be written as the product of an integer with itself. The name comes from geometry: n² is exactly the area of a square whose side has length n.

• Formula: n² = n × n
• First squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
• Example: 12² = 12 × 12 = 144
• Sign: n² is always non-negative, even when n is negative (e.g., (−5)² = 25).

Square Numbers Table (1 to 100)

Complete table of n² for integers from 1 to 100. Useful for quick lookup, mental math, and verification.

Properties and Patterns of Square Numbers

Square numbers have a rich set of patterns that make them easier to recognize, compute, and reason about.

• Sum of odd numbers: n² equals the sum of the first n odd numbers. For example, 5² = 1 + 3 + 5 + 7 + 9 = 25.
• Consecutive squares: (n + 1)² − n² = 2n + 1, which is always an odd number. So the gaps between consecutive squares are 3, 5, 7, 9, 11, … (odd numbers).
• Last digits: the last digit of any square is always one of 0, 1, 4, 5, 6, or 9. Squares never end in 2, 3, 7, or 8.
• Even and odd: the square of an even number is even; the square of an odd number is odd.
• Modulo 4: every square is congruent to either 0 or 1 (mod 4). This is a useful test for whether a number can be a perfect square.

Mental Math Shortcuts for Squaring

Several shortcuts let you compute squares quickly without a calculator.

Numbers ending in 5

To square a number ending in 5, take the tens digit a, multiply it by a + 1, and append 25.

• 25² → 2 × 3 = 6, append 25 → 625
• 35² → 3 × 4 = 12, append 25 → 1,225
• 85² → 8 × 9 = 72, append 25 → 7,225

Near a known square

Use the identity (n ± k)² = n² ± 2nk + k². If you know 30² = 900, then:

• 31² = 30² + 2(30)(1) + 1 = 900 + 60 + 1 = 961
• 29² = 30² − 2(30)(1) + 1 = 900 − 60 + 1 = 841

Difference of squares

Use n² − m² = (n − m)(n + m). For example, 51² − 49² = (2)(100) = 200, so 51² = 49² + 200 = 2,401 + 200 = 2,601.

Applications of Square Numbers

Geometry

• Area of a square: a square with side length n has area n².
• Pythagorean theorem: a² + b² = c² for right triangles.
• Distance formula: the squared distance between (x₁, y₁) and (x₂, y₂) is (x₂ − x₁)² + (y₂ − y₁)².

Algebra

• Quadratic equations: ax² + bx + c = 0 uses the square term as its highest degree.
• Completing the square: a key technique for solving quadratics and deriving the quadratic formula.
• Binomial expansion: (a + b)² = a² + 2ab + b².

Number Theory

• Sum of two squares theorem: a positive integer is the sum of two squares if and only if every prime factor of the form 4k + 3 occurs an even number of times.
• Lagrange's four-square theorem: every positive integer is the sum of at most four squares.

Statistics

• Variance and standard deviation: built from squared deviations from the mean.
• Least-squares regression: minimizes the sum of squared errors.

Frequently Asked Questions

What is a square number in simple terms?

A square number is the result of multiplying an integer by itself. For example, 36 is a square number because 36 = 6 × 6.

Is 0 a square number?

Yes. By definition, 0² = 0 × 0 = 0, so 0 is a perfect square.

Is 1 a square number?

Yes. 1² = 1, so 1 is a perfect square.

How can I tell quickly if a number is a perfect square?

Check the last digit first — if it ends in 2, 3, 7, or 8, it cannot be a perfect square. Otherwise, estimate its square root and check whether rounding gives an integer.

What is the difference between square numbers and squared numbers?

"Square number" usually refers specifically to perfect squares (squares of integers). "Squared number" is more general and can mean the square of any real number, including non-integers like 2.5² = 6.25.

References

• Khan Academy — Introduction to exponents and squaring.
• Wikipedia — Square number (definition, properties, and theorems).
• OEIS A000290 — The sequence of square numbers.

See also