Finding the square of a four-digit number can seem daunting, but there are some helpful tricks that make this process quicker and easier. One of the technique is the base method, where you select a base number close to the four-digit number you're squaring. For example, squaring 104 can be done using a base of 100. By applying this method, you reduce complex multiplication into simpler steps, making the calculation much faster.
In this article, we will discuss the trick in detail.
What is Square of a Number?
The square of a number is the result of multiplying that number by itself. In mathematical terms, if you have a number xxx, its square is represented as x2x^2x2, which is calculated as:
x2 = x × x
For example:
- The square of 2 is 22 = 2 × 2 = 42.
- The square of 5 is 52 = 5 × 5 = 52.
Note: In general, squaring a number gives the area of a square with sides of length equal to that number.
Why it is Hard to Find Square of Larger Numbers?
Finding the square of larger numbers is harder because it involves more complex multiplication with multiple digits, increasing the chance of errors. The size of the results grows rapidly, making manual calculations difficult to manage mentally.
Trick to Find the Square of a 4-Digit Number
The trick to squaring a 4-digit number involves a few easy-to-follow steps. Let’s take an example and go through the process step by step:
Example: Find the square of 1024.
Solution:
Step 1: Split the number
Split the number into two parts:
The first part consists of the first two digits and remaining zeros: 1000.
The second part is the last two digits: 24.
So, we have 1024 as 1000 + 24.
Step 2: Square each part
Now, square each of the two parts separately:
10002 = 1000000
242 = 576
Step 3: Multiply the parts and double the result
Multiply the first part by the second part, and then double the result:
1000 × 24 = 24000
Double it: 24000 × 2 = 48000
Step 4: Combine the results
Now, add all the results together. First, add the square of the first part and the square of the second part:
1000000 + 576 + 48000 = 1048576
Thus, the square of 1024 is 1048576.
Why this Works?
The formula used in this trick is based on the algebraic identity:
(A + B)2 = A2 + B2 + 2AB
Here:
A = the first part of the number (e.g., the first two digits of a 4-digit number).
B = the second part of the number (e.g., the last two digits).
Solved Examples
Example 1: Square of 1035.
Solution:
Split the number:
A = 1000, B = 35
Square each part:
1002 = 1,000,000
352 = 1225
Multiply and double:
1000 × 35 = 35,000
35,000 × 2 = 70,000
Add all parts:
1,000,000+70,000+1225 = 1,071,225
Answer: 10352 = 1,071,225
Example 2: Square of 995.
Solution:
Base: 1000
Difference from base: 1000−995 = 5
Calculate the square of difference: 52 = 25
Subtract from base: 1000−5 = 995
Final result: 995 × 1000−25 = 995,000−25 = 990,025
Answer: 9952 = 990,025
Example 3: Square of 1201.
Solution:
Split the number:
A = 1200,
B = 1
Square each part:
12002 = 1,440,000
12 = 1
Multiply and double:
1200 × 1 = 1200
1200 × 2 = 2400
Add all parts:
1,440,000+2400+1 = 1,442,401
Answer: 12012 = 1,442,401
Example 4: Square of 1018.
Solution:
Base: 1000
Difference from base: 1018−1000 = 18
Calculate square of difference: 182 = 324
Add to the base: 1018+18 = 1036
Multiply by base: 1036 × 1000 = 1,036,000
Add the squared difference: 1,036,000+324 = 1,036,324
Answer: 10182 = 1,036,324
Example 5: Square of 998.
Solution:
A = 1000,
B = −2
Square each part:
10002 = 1,000,000
(-2)2 = 4
Multiply and double:
1000 × −2 = −2000
−2000 × 2 = −4000
Add all parts:
1,000,000−4000+4 = 996,004
Answer: 9982 = 996,004
Example 6: Square of 1050.
Solution:
Base: 1000
Difference from base: 1050−1000 = 50
Calculate the square of difference: 502 = 2500
Add to the base: 1050+50 = 1100
Multiply by base: 1100 × 1000 = 1,100,000
Add the squared difference: 1,100,000+2500 = 1,102,500
Answer: 10502 = 1,102,500
Example 7: Square of 1234.
Solution:
Split the number:
A = 1200,
B = 34
Square each part:
12002 = 1,440,000
342 = 1156
Multiply and double:
1200 × 34 = 40,800
40,800 × 2 = 81,600
Add all parts:
1,440,000+81,600+1156 = 1,522,756
Answer: 12342 = 1,522,756
Practice Problems
- 10252
- 10022
- 11952
- 9972
- 11042
- 10432
- 9802
- 10672
- 11322
- 11502
Answer Key
- 1,050,625
- 1,004,004
- 1,428,025
- 994,009
- 1,219,216
- 1,088,849
- 960,400
- 1,138,489
- 1,281,424
- 1,322,500
Conclusion
This squaring trick is a valuable tool for students looking to improve their mental math skills and speed in competitive exams. Using simple algebraic formulas and breaking down a 4-digit number into still smaller parts, we find that the final calculations are relatively simple and sure. When you master this trick you get to improve not only your exam performance but even your general mathematic skills.
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