Square of a Number
Definition
The square of a number is obtained by multiplying the number by itself.
Square of a = a × a = a²
Visual Representation
If a number represents the side of a square, then squaring it gives the area of that square. For example, a square with side 5 units has an area of:
Area = 5² = 25 square units
Examples
- 4² = 16
- (−3)² = 9
- 0² = 0
Properties
- The square of a positive number is positive.
- The square of a negative number is positive.
- The square of zero is zero.
- The square of an integer is always a non-negative integer.
List of Perfect Squares (1–30)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900
Square Root of a Number
Definition
The square root of a number is a value which, when multiplied by itself, yields the original number.
√a = b ⇒ b² = a
Types of Square Roots
- Exact (Perfect Square Roots): √49 = 7
- Irrational (Non-Perfect Square Roots): √2 ≈ 1.414
Symbol
The radical symbol √ is used to denote the square root.
Examples
- √25 = 5
- √0 = 0
- √(−4) is undefined in real numbers (it's 2i in complex numbers)
Properties
- Square root of a number is always non-negative in real numbers.
- √a² = |a|
Methods to Find Square Roots
a) Prime Factorization Method
Useful for perfect squares. Factor the number into primes, pair the same primes.
Example: √144 = √(2⁴ × 3²) = 2² × 3 = 12
b) Long Division Method
Used for larger numbers or numbers with decimals.
Steps:
- Pair the digits from right to left.
- Divide, find the closest square, subtract, bring down next pair.
- Repeat to get more decimal places.
Example: Find √625 using this method:
√625 = 25
c) Estimation and Approximation
Identify two closest perfect squares and estimate between them.
Example: √10 ≈ 3.16 (between √9 = 3 and √16 = 4)
Advanced Concepts
Square Roots of Decimals
To find √0.64, treat as √64/10 = 0.8
Square Roots of Fractions
√(a/b) = √a / √b
Example: √(25/49) = 5/7
Irrational Square Roots
These cannot be expressed exactly as fractions and have non-terminating, non-repeating decimals.
Examples: √2, √3, √5
Common Mistakes to Avoid
- Assuming √a² = a without considering |a|
- Forgetting that √(−a) is not real for positive a
- Incorrect pairing in long division method
- Applying square root rules to sums or differences: √(a + b) ≠ √a + √b
Real-Life Applications
- Geometry: Calculating distances, areas
- Physics: Computing energy, motion equations
- Engineering: Stress analysis, optimization
- Finance: Risk measurement, standard deviation
- Computer Science: Algorithms like Euclidean distance
Practice Problems
Basic
- Find the square of: 11, -8, 0
- Determine if the following are perfect squares: 81, 72, 144
- Find the square roots of: 256, 169, 625
Intermediate
- Estimate √45 to two decimal places.
- Use long division method to find √324
- Simplify √(16/81)
Advanced
- Solve x² = 225
- Find the square root of 0.0025
- Prove √(a² + b²) ≥ √a² + √b² is false with a counterexample.
8. Solutions
- 11² = 121, (−8)² = 64, 0² = 0
- 81 and 144 are perfect squares; 72 is not.
- √256 = 16, √169 = 13, √625 = 25
- √45 ≈ 6.71
- √324 = 18
- √(16/81) = 4/9
- x = ±15
- √0.0025 = 0.05
- a = 1, b = 1 ⇒ √2 ≈ 1.41 < 1 + 1 = 2
Conclusion
Understanding the concepts of squares and square roots builds a solid foundation for advanced mathematics. Whether it is solving equations, analyzing data, or applying geometry, these concepts appear in many real-world and academic scenarios. Regular practice and correct application are key to mastery.