Tricks and Formulas to Calculate Squares Without Multiplication
The concept of finding the square of a number is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Knowing how to quickly and accurately square numbers is especially useful for board exams, competitive tests, and everyday calculations.
Understanding Finding the Square of a Number
A square of a number refers to the result you get when multiplying a number by itself. For example, the square of 7 is \( 7 \times 7 = 49 \). This concept is widely used in algebraic identities, geometry, and arithmetic operations.
Formula Used in Finding the Square of a Number
The standard formula to find the square of a number "n" is: \( \text{Square of } n = n \times n \) or \( n^2 \).
Here’s a helpful table to understand finding the square of a number more clearly:
Squares of Numbers Table (1-10)
| Number | Square | Perfect Square? |
|---|---|---|
| 1 | 1 | Yes |
| 2 | 4 | Yes |
| 3 | 9 | Yes |
| 4 | 16 | Yes |
| 5 | 25 | Yes |
| 6 | 36 | Yes |
| 7 | 49 | Yes |
| 8 | 64 | Yes |
| 9 | 81 | Yes |
| 10 | 100 | Yes |
This table shows how squaring creates a special set of numbers called perfect squares.
Worked Example – Solving a Square Calculation
Let's take a step-by-step approach to finding the square of 32:
1. Write the number as a sum for easier calculation: \( 32 = 30 + 2 \ )2. Apply the formula \( (a+b)^2 = a^2 + 2ab + b^2 \ ):
3. Calculate each part:
\( 2ab = 2 \times 30 \times 2 = 120 \)
\( b^2 = 2^2 = 4 \)
4. Add all parts together:
Final answer: The square of 32 is 1024.
Shortcuts and Patterns for Finding Squares
For numbers ending with 5, you can use a simple trick. If a number is n5 (like 25, 35, 75), its square is always \( n \times (n+1) \) followed by 25.
Example: Square of 65
1. Remove the 5: \( n = 6 \)2. Multiply n by (n+1): \( 6 \times 7 = 42 \)
3. Write 25 at the end: 4225
So, \( 65^2 = 4225 \).
Finding Square of a Number Without Actual Multiplication
Algebraic identities like \( (a+b)^2 \), patterns, and tables can help avoid direct multiplication. For instance, squaring numbers close to 100 or 1000 can be much quicker with patterns and identities.
Square of 98:
1. \( 98 = 100 - 2 \)2. Use identity: \( (a-b)^2 = a^2 - 2ab + b^2 \)
\( 2ab = 2 \times 100 \times 2 = 400 \)
\( b^2 = 2^2 = 4 \)
3. Now, \( 10000 - 400 + 4 = 9604 \)
So, the square of 98 is 9604.
Using Programming to Find the Square of a Number
You can also use simple programs to find the square. Here is an example in Python:
n = 17
square = n * n
print("The square of", n, "is", square)
This will output: The square of 17 is 289
Difference: Square and Square Root
| Operation | Example | Result |
|---|---|---|
| Finding Square | 8 | \( 8 \times 8 = 64 \) |
| Finding Square Root | 64 | \( \sqrt{64} = 8 \) |
Squaring is raising a number to the power of 2. Square root is finding the original number whose square gives the answer.
Practice Problems
- Find the square of 15, 21, and 99 using the steps above.
- Use the shortcut to square 45 and 85.
- Which numbers between 40 and 50 are perfect squares?
- Find the unit digit of the square of 54.
Common Mistakes to Avoid
- Confusing square with square root.
- Forgetting to multiply the number by itself (not by 2).
- Missing steps in applying the formula or shortcut.
Real-World Applications
The concept of finding the square of a number is found in calculating areas (such as finding the area of a square), physics equations, quadratic equations in algebra, and in many competitive exams. Practice with Vedantu helps to quickly master this important skill for school, exams, and life beyond the classroom.
We explored the idea of finding the square of a number, how to apply formulas, use shortcut patterns, and understand where squares appear in real-life problems. Strengthen your maths skills by practicing similar questions, and explore more techniques with Vedantu for higher confidence and accuracy.
For further learning, check out these helpful pages:
FAQs on How to Find the Square of a Number Easily
1. What is finding the square of a number?
Finding the **square of a number** means multiplying the number by itself. Mathematically, if the number is x, then its square is x × x or x². Squaring is a fundamental concept in **arithmetic**, **algebra**, and **geometry** and is frequently used in **board exams** and other competitive tests.
2. What is the formula to find the square of a number?
The standard formula to find the square of a number x is: Square = x × x = x². Additionally, algebraic identities such as (a + b)² = a² + 2ab + b² can be used to calculate squares of numbers broken into parts for easier computation.
3. What is the shortcut to find the square of a number?
Shortcuts to find the square of a number include the use of **algebraic identities** and pattern-based tricks such as:
• For numbers ending with 5: (n5)² = n × (n + 1) hundred + 25
• Using the formula (a + b)² = a² + 2ab + b² to simplify calculations
• Recognizing perfect squares and applying mental math techniques for faster results, important for quick answers during exams.
4. How to find the square of a number in Python or Java?
In programming languages like Python and Java, the square of a number can be found by multiplying the number by itself:
• In Python: square = number * number or square = number ** 2
• In Java: int square = number * number;
This helps students learn **coding** implementations related to mathematical concepts.
5. How to find the square of a number without actual multiplication?
To find the square without direct multiplication, you can use:
• **Algebraic identities** like (a + b)² = a² + 2ab + b²
• **Pattern methods**, especially for numbers ending with 5 using (n5)² = n × (n + 1) hundred + 25
• Applying **mental math tricks** and breaking numbers into parts for easier summation, useful for faster calculations during board exams.
6. What is the square of 32 or 17?
The square of 32 is calculated as 32 × 32 = 1024. The square of 17 is 17 × 17 = 289. These examples demonstrate expanding using (a + b)² formulas or direct multiplication and can be memorized as part of important **perfect squares**.
7. What is the difference between square and square root?
The **square** of a number is the result of multiplying the number by itself. The **square root** of a number is the inverse operation—it gives the original number whose square equals the given number. For example, the square of 5 is 25, and the square root of 25 is 5. Understanding this distinction is crucial to avoid errors in **algebraic calculations** and exam problems.
8. What are Pythagorean triplets and how are they related to squares?
**Pythagorean triplets** are sets of three natural numbers (a, b, c) that satisfy the relation a² + b² = c², representing the sides of a right triangle. Examples include (3, 4, 5) and (6, 8, 10). These triplets demonstrate the practical use of squares in **geometry** and help students recognize perfect squares and right-angle properties.
9. How to check if a number is a perfect square?
A number is a **perfect square** if its square root is an integer. You can check this by:
• Calculating the square root and verifying if it is whole
• Memorizing common perfect squares such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
• Using calculators or programming methods to confirm
This skill helps in **simplifying radical expressions** and solving algebraic problems efficiently.
10. Why is learning to find squares important for exams and daily life?
Learning to find squares is vital because:
• It forms the basis for solving many **algebra** and **geometry** problems
• It enhances **mental math skills** useful for quick calculations
• It is frequently tested in **board exams** and **competitive tests**
• It helps in understanding more complex concepts like **quadratic equations** and **Pythagoras theorem**
Mastering squares leads to better speed and accuracy in mathematics.
11. What are common mistakes students make while calculating squares?
Common mistakes include:
• Forgetting to multiply the number by itself
• Confusing **square** with **square root**
• Missing out on carrying digits in manual multiplication
• Incorrectly applying shortcuts without understanding rules
• Neglecting place value when expanding numbers
Being aware of these mistakes helps in avoiding errors and scoring higher marks in exams.
12. How can pattern recognition help in finding squares quickly?
Recognizing patterns such as:
• Squares of numbers ending in 5
• Relationship between consecutive squares
• Using algebraic identities like (a + b)²
helps in mentally calculating squares faster without full multiplication. This improves exam performance and builds strong **number sense** in students.
