Squares and Square Roots

Patterns in Square Numbers

Math Patterns: Suppose you and your curious friends are talking about square numbers. One of them wants to know about the patterns among the square numbers. Since now you are familiar with the patterns of odd and even numbers. You are also familiar with the corresponding ending (unit’s place) digits of the square numbers of the respective numbers.

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But what are the other patterns associated with square numbers? In this section, we will learn about the math patterns associated with and in between square numbers. Let’s us discuss some of these math patterns in the following article.

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Math Patterns Between Square Numbers

Difference Between Square Numbers

The difference between consecutive squares is always an odd number. If n is any natural number then the difference between the square of n and the square of the next natural number, (n + 1) is 2n + 1.

Adding Triangular Numbers

Drawing number into triangular dot pattern and making a square (with same numbers of dots in rows and columns) will give a square number.

Math Patterns

Similarly, 16 = 42 = 32 + (3 + 3 + 1), 25 = 52 = 42 + (4 + 4 + 1) and so on.

Adding Odd Numbers

A number will be a square number if it is the sum of consecutive or successive odd numbers starting from 1. Let us now consider,

1 (first odd number) = 1 = 12
1 + 3 (sum of first two odd numbers) = 4 = 22
1 + 3 + 5 (sum of first three odd numbers) = 9 = 32
1 + 3 + 5 + 7 (sum of first four odd numbers) = 16 = 42
1 + 3 + 5 + 7 + 9 (sum of first five odd numbers) = 25 = 52

A number which is not a perfect square cannot be put in this form. Also, we can check if a number is a perfect square or not. Let us now consider the number 16. Subtracting 16 from the series of successive odd numbers and getting 0, as a result, will tell that 16 is a perfect square.

16 – 1 = 15
15 − 3 = 12
12 − 5 = 7
7 − 7 = 0

This means that 16 is a perfect square. Any non-perfect square will leave a remainder other than zero. Try out the successive subtraction by consecutive odd numbers for 35.

Sum of Consecutive Natural Numbers

Any odd square number can be expressed as the sum of two consecutive natural numbers. Let us now take any odd perfect square say 441, 212 = 441 = 220 + 221. Similarly, we have 232 = 529 = 264 + 265 and so on.

Product of Two Consecutive Even Or Odd Natural Numbers

Consider the product of two consecutive even numbers For Example, 20 × 22 = (21 – 1) × (21 + 1) = 212 – 1. The product of two consecutive odd numbers, 21 × 23 = (22 – 1) × (22 + 1) = 222 – 1. Give rise to an identity

(a + 1) (a – 1) = a– 1.

Solved Examples for You

Problem: Check if 121 is a perfect square by the method of the sum of successive odd numbers from the math patterns.

Solution: Subtracting the number by a series of successive odd numbers will give

121 – 1 = 120
120 – 3 = 117
117 – 5 = 112
112 – 7 = 105
105 – 9 = 96
96 – 11 = 85
85 – 13 = 72
72 – 15 = 57
57 – 17 = 40
40 – 19 = 21
21 – 21 = 0

So as we are getting zero in the end hence 121 is a perfect square.

Problem: Express 3721 as the sum of consecutive natural numbers.

Solution: we know 612 = 3721. Taking the half of 3721 we can have an estimation for the choice of the consecutive natural numbers. So, 3721 can be expressed as the sum of 1860 and 1861 i.e., 3721 = 1860 + 1861.

Problem: A number has 1  zero at the end, the square of that number will have _____.

  1. 1 Zero
  2. 2 Zeros
  3. 3 Zeros
  4. 4 Zeros

Solution: The correct option is B. For example,

  • 10 ×  10 = 100
  • 20 × 20  = 400

Therefore D is the correct option for the problem.

Solved Questions for You

Question 1: Why do we study patterns in mathematics?

Answer: The skill to identify and form patterns assists us in making predictions on the basis of our observations. Understanding patterns help prepare kids for learning complex number concepts and mathematical operations. Moreover, patterns permit us to see relationships and form generalizations.

Question 2: What are the types of patterns in math?

Answer: A pattern refers to a series or sequence which repeats. There are two major kinds of math patterns. They are number patterns or sequences of numbers. These are arranged as per a rule or rules. Then, we have shape patterns that are tagged by making use of letters and the way that they repeat.

Question 3: What is the difference between Square Numbers?

Answer: The difference between consecutive squares is constantly an odd number. If n is any natural number subsequently the difference between the square of n and the square of the next natural number, (n + 1) will be 2n + 1.

Question 4: What is a visual pattern?

Answer: A visual pattern refers to a sequence of pictures or geometric objects which have been formed on the basis of some rule. The sequence of pictures that keep repeating is referred to as the pattern unit. Some visual patterns are on the basis of a picture or geometric object which keeps increasing or decreasing in size in a particular manner.

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