Properties of Squares and Square Roots

Definition of a Square

• A square is a number that can be expressed as the product of some integer multiplied by itself (e.g., 4 = 2 × 2)
• Notation: a² (a squared)

Properties of Squares

• A square is always non-negative (≥ 0)
• The square of an even number is always even
• The square of an odd number is always odd
• Squares of consecutive integers differ by an odd number

Square Roots

• A square root of a number is a value that, when multiplied by itself, gives the original number
• Notation: √a (square root of a)
• Every positive number has two square roots: a positive square root and a negative square root (e.g., √16 = 4 or -4)
• The positive square root is denoted by the symbol √ (e.g., √16 = 4)

Calculating Square Roots

• Perfect squares: can be calculated exactly (e.g., √16 = 4)
• Non-perfect squares: can be approximated using numerical methods (e.g., √2 ≈ 1.414)

Real-World Applications

• Area of a square: side² (e.g., area of a square with side 4 cm = 4² = 16 cm²)
• Distance and Pythagorean theorem: a² + b² = c² (e.g., distance between two points in a coordinate plane)
• Geometry and trigonometry: squares and square roots are used to calculate lengths, areas, and angles of shapes.

Definition of a Square

• A square is a number that can be expressed as the product of some integer multiplied by itself, such as 4 = 2 × 2.
• The notation for a square is a², read as "a squared".

Properties of Squares

• A square is always non-negative, meaning it is greater than or equal to 0.
• The square of an even number is always even.
• The square of an odd number is always odd.
• The difference between squares of consecutive integers is always an odd number.

Square Roots

• A square root of a number is a value that, when multiplied by itself, gives the original number.
• The notation for a square root is √a, read as "square root of a".
• Every positive number has two square roots: a positive square root and a negative square root.
• The positive square root is denoted by the symbol √, such as √16 = 4.

Calculating Square Roots

• Perfect squares can be calculated exactly, such as √16 = 4.
• Non-perfect squares can be approximated using numerical methods, such as √2 ≈ 1.414.

Real-World Applications

• The area of a square can be calculated using the formula: side², such as the area of a square with a side of 4 cm is 4² = 16 cm².
• The distance and Pythagorean theorem can be calculated using the formula: a² + b² = c², such as the distance between two points in a coordinate plane.
• Geometry and trigonometry uses squares and square roots to calculate lengths, areas, and angles of shapes.