Squares of Numbers
The square of a number is the result of multiplying a number by itself. If n is a number, then n × n = n² is called the square of n.
Squares are one of the most fundamental operations in mathematics. They appear in area calculations, the Pythagoras theorem, algebraic identities, and physics formulas.
A perfect square is a number that is the square of a whole number. For example, 1, 4, 9, 16, 25, 36, ... are perfect squares because they equal 1², 2², 3², 4², 5², 6², ... respectively.
What is Squares of Numbers?
Definition: The square of a number n is the product of the number multiplied by itself.
n² = n × n
Where:
- n is any number (integer, fraction, or decimal)
- n² is read as "n squared" or "n to the power 2"
- The result is always non-negative (≥ 0)
Definition: A perfect square (or square number) is a natural number that can be expressed as the square of a natural number.
- The first 20 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Methods
Methods for finding squares:
Method 1: Direct Multiplication
- Simply multiply the number by itself.
- Example: 13² = 13 × 13 = 169
Method 2: Using the identity (a + b)² = a² + 2ab + b²
- Split the number into two parts: a + b.
- Apply the identity.
- Example: 23² = (20 + 3)² = 400 + 120 + 9 = 529
Method 3: Using the pattern for numbers ending in 5
- For a number like n5 (e.g., 25, 35, 45):
- Multiply n by (n + 1). Write the result.
- Append 25 at the end.
- Example: 35² → 3 × 4 = 12, append 25 → 1225
Method 4: Using consecutive square difference
- n² = (n − 1)² + (n − 1) + n
- Or equivalently: n² − (n − 1)² = 2n − 1
- This means the difference between consecutive perfect squares is always an odd number.
Solved Examples
Example 1: Example 1: Square of a two-digit number
Problem: Find 14².
Solution:
Using direct multiplication:
- 14² = 14 × 14
- = 14 × 10 + 14 × 4
- = 140 + 56
- = 196
Answer: 14² = 196
Example 2: Example 2: Using (a + b)² identity
Problem: Find 27² using the algebraic identity.
Solution:
Given:
- 27 = 20 + 7, so a = 20, b = 7
Using (a + b)² = a² + 2ab + b²:
- = 20² + 2(20)(7) + 7²
- = 400 + 280 + 49
- = 729
Answer: 27² = 729
Example 3: Example 3: Number ending in 5
Problem: Find 45² using the shortcut for numbers ending in 5.
Solution:
Steps:
- Number = 45. The digit before 5 is 4.
- Multiply 4 by (4 + 1) = 4 × 5 = 20
- Append 25: 2025
Check: 45 × 45 = 2025. Correct.
Answer: 45² = 2025
Example 4: Example 4: Square of a negative number
Problem: Find (−8)².
Solution:
Steps:
- (−8)² = (−8) × (−8)
- Negative × Negative = Positive
- = 64
Important: (−8)² = 64, but −8² = −(8²) = −64. The parentheses matter!
Answer: (−8)² = 64
Example 5: Example 5: Is 2025 a perfect square?
Problem: Check whether 2025 is a perfect square.
Solution:
Steps:
- Find the prime factorisation: 2025 = 3 × 675 = 3 × 3 × 225 = 3² × 225
- 225 = 15² = (3 × 5)² = 3² × 5²
- So 2025 = 3² × 3² × 5² = 3⁴ × 5²
- All prime factors have even powers
- √2025 = 3² × 5 = 9 × 5 = 45
Answer: Yes, 2025 is a perfect square. 2025 = 45².
Example 6: Example 6: Sum of first n odd numbers
Problem: Find the sum: 1 + 3 + 5 + 7 + 9 + 11.
Solution:
Using the property: Sum of first n odd numbers = n²
- Count the terms: there are 6 odd numbers
- Sum = 6² = 36
Check: 1 + 3 + 5 + 7 + 9 + 11 = 36. Correct.
Answer: Sum = 36
Example 7: Example 7: Square of a fraction
Problem: Find (3/4)².
Solution:
Steps:
- (3/4)² = 3²/4²
- = 9/16
Answer: (3/4)² = 9/16
Example 8: Example 8: Square of a decimal
Problem: Find (0.6)².
Solution:
Steps:
- (0.6)² = 0.6 × 0.6
- = 0.36
Alternatively: 0.6 = 6/10, so (6/10)² = 36/100 = 0.36
Answer: (0.6)² = 0.36
Example 9: Example 9: Using difference of consecutive squares
Problem: Given that 30² = 900, find 31² without multiplying.
Solution:
Using the property: n² = (n − 1)² + 2n − 1
- 31² = 30² + 2(31) − 1
- = 900 + 62 − 1
- = 900 + 61
- = 961
Check: 31 × 31 = 961. Correct.
Answer: 31² = 961
Example 10: Example 10: Can a number ending in 7 be a perfect square?
Problem: Without calculating, determine whether 4587 can be a perfect square.
Solution:
Steps:
- The number 4587 ends in 7.
- A perfect square can only end in 0, 1, 4, 5, 6, or 9.
- Since 4587 ends in 7, it cannot be a perfect square.
Answer: No, 4587 is not a perfect square.
Real-World Applications
Real-world applications of squares:
- Area calculation: Area of a square with side a = a². A square room of side 5 m has area 25 m².
- Pythagoras theorem: In a right triangle, c² = a² + b². Finding distances uses squares.
- Physics: Kinetic energy = ½mv². The square of velocity determines energy.
- Statistics: Standard deviation uses squares of deviations from the mean.
- Computer science: Checking if a number is a perfect square is a common algorithmic problem.
- Finance: Compound interest formulas involve squaring the growth factor for 2 years.
Key Points to Remember
- The square of n is n² = n × n.
- The square of any number is always non-negative (≥ 0).
- (−n)² = n². Squaring a negative gives a positive.
- Perfect squares end in 0, 1, 4, 5, 6, or 9 only. Never in 2, 3, 7, or 8.
- Square of even number = even. Square of odd number = odd.
- Difference of consecutive squares: n² − (n−1)² = 2n − 1.
- Sum of first n odd numbers = n².
- For numbers ending in 5: (n5)² = n(n+1) followed by 25.
- A perfect square has an odd number of factors.
- In the prime factorisation of a perfect square, every prime factor appears an even number of times.
Practice Problems
- Find 18².
- Find 65² using the shortcut for numbers ending in 5.
- Using the identity (a + b)², find 32².
- Is 1764 a perfect square? If yes, find its square root.
- Without calculating, determine if 3378 can be a perfect square.
- Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15.
- Given 25² = 625, find 26² without multiplying.
- Find (−12)² and (−0.7)².
Frequently Asked Questions
Q1. What is the square of a number?
The square of a number n is the product n × n, written as n². For example, the square of 7 is 7² = 49.
Q2. What is a perfect square?
A perfect square is a natural number that equals the square of some natural number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Q3. Can a negative number be a perfect square?
No. Since squaring any real number gives a non-negative result, perfect squares are always ≥ 0. A number like −4 is not a perfect square.
Q4. What is the difference between (−5)² and −5²?
(−5)² = (−5) × (−5) = 25 (the square of negative 5). But −5² = −(5²) = −25 (the negative of 5 squared). The parentheses make a crucial difference.
Q5. How can you tell if a number is NOT a perfect square without calculating?
Check the last digit. If a number ends in 2, 3, 7, or 8, it is NOT a perfect square. Perfect squares can only end in 0, 1, 4, 5, 6, or 9.
Q6. What is the shortcut for squaring numbers ending in 5?
For a number like n5: multiply n by (n + 1) and append 25. Example: 75² → 7 × 8 = 56, append 25 → 5625.
Q7. Why is the sum of first n odd numbers equal to n²?
This can be visualised with dots arranged in an L-shape pattern. Each successive odd number adds another L-shaped layer to a square arrangement, making the total a perfect square.
Q8. How do you square a fraction?
Square the numerator and denominator separately: (a/b)² = a²/b². For example, (2/3)² = 4/9.
Q9. What is the relationship between consecutive squares?
The difference between n² and (n−1)² is always 2n − 1 (an odd number). For example, 10² − 9² = 100 − 81 = 19 = 2(10) − 1.
Q10. How many perfect squares are there between 1 and 100?
There are 10 perfect squares between 1 and 100 (inclusive): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are 1² through 10².
