Squares and Square Roots

Formation of Squares Using Patterns

Number Pattern: Suppose you have to find the square of a big number. How can you find it? Can you answer this? Your answer will be by multiplying the number by itself. But isn’t it time-consuming and tedious to find which number will give that square? Calculating each and every square before getting the right answer is not an easy task.

Can we associate any patterns with the formation of squares? Are there any number of patterns associated with the formation of squares? Let us now learn about these number pattern in this following article.

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Formation of Squares Using Number Pattern

Finding the squares of small numbers are quite easy to find. But is it the same for more than a single digit number? The answer is not every time. Finding the square of 20 is quite easy as compared to 29. Also, calculating the square of 400 is easy but that of 54 is not. Let us find some patterns for finding the squares. Let us calculate the square of a number without actual multiplication.

Without Actual Multiplication

Consider calculating the square of 54. We know 54 = 50 + 4. So, 542 = (50 + 4)2 = (50 + 4) (50 + 4) = 50(50 + 4) + 4(50 + 4) = 502 + (50 × 4) + (4 × 50) + 42 = 2500 + 200 + 200 + 16 = 2916. This is similar to the identity (a + b)2 = a2 + b2 + 2ab.

Other Number Pattern in Squares

This pattern is associated with finding the square of the numbers having 5 in their unit’s place. For finding the square we have to multiply the number(s) except for those in unit’s place with the next coming number in hundreds and add 25.

Consider squaring 65. Now,  652 = 4225 = (6 × 7) hundreds + 25 = 4200 + 25 = 4225. Similarly, 1152 = 13225 = (11 × 12) hundreds + 25 = 13200 + 25 = 13225.

Pythagorean Triplets

There are some cases for which the sum of the squares of two numbers will give rise to the square of another number. The three numbers are the Pythagorean triplets. The examples of Pythagorean triplets are 3, 4 and 5; 5, 12 and 13; 6, 8 and 10 etc.

  • 32 + 42 = 9 + 16 = 25 = 52.
  • 52 + 122 = 25 + 144 = 169 = 132.

This gives rise to the Pythagoras Theorem in a right-angled triangle.

Number Pattern

But the real question is how can we find such triplets? Any natural number, n> 1 which satisfies the condition (2n) + (n2 – 1) 2 = (n2 + 1) 2 satisfies the condition for Pythagorean triplets. Here, 2n, n2 – 1, and n2 + 1 forms a Pythagorean triplet.

Solved Examples for You

Question 1:  Write a Pythagorean triplet in which 65 is the largest number.

Answer : The general form for getting Pythagorean triplets is 2n, n2 – 1, and n2 + 1. Since 65 is the largest number so n2 + 1 = 65 = n2 = 64 or n = 8. We get n2 – 1 = 64 – 1 = 63. And 2n = 2 × 8 = 16. The triplets are 16, 63, and 65 with 65 as the largest number.

Question 2:  Find the square of 89 without actual multiplication.

Answer : The number 89 can be written as 89 = 90 – 1. Therefore, 892 = (90 – 1)2 = (90 – 1) (90 – 1) = 90(90 – 1) – 1(90 – 1) = 902 – (90 × 1 – (1 × 90 + 12 = 8100 – 90 – 90 + 1 = 7921. This is similar to the identity (a – b)2 = a2 – 2ab + b2.

Question 3:  When we combine two consecutive triangular numbers, we get a __________.

  1. Square Number
  2. Consecutive Number
  3. Non-Square number
  4. Zero

Answer : The correct option is A. Triangular numbers are obtained by continued summation of natural numbers like 1, 2, 3, 4, 5, 6…..Therefore the set of triangular number is 1, 3, 6, 10, 15…When we add two triangular numbers we get a square number, 1 + 3 = 4 = 2², 3 + 6 = 9 = 3²  and so on…. Therefore A is the correct option.

Question 4: What is a pattern rule?

Answer: An explicit pattern rule refers to a pattern rule that tells us how to obtain any term in the pattern without having to list all the terms prior to it. For instance, an explicit pattern rule for 3, 6, 9, 12 uses the first term (3) and the common difference (3).

Question 5: What is the formula for number pattern?

Answer: A linear number pattern refers to a list of numbers where the difference between each number in the list is the same. The formula for the nth term of a linear number pattern indicated an, is an = dn – c. Over here, d is the common difference in the linear pattern and c is a constant number.

Question 6: What is a formula in math?

Answer: Formula refers to a group of mathematical symbols which state a relationship or that we use to solve a problem or a way to make something. In other words, a group of math symbols expressing the relationship between the circumference of a circle and the diameter is an example of a formula.

Question 7: Is there a pattern to random numbers?

Answer: Some or most of the “random” numbers are not actually random. They may follow subtle patterns that people observe over long periods of time, or over many instances of producing random numbers.

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