Expressing Square Numbers as Sum of consecutive Natural Numbers

You can express any odd perfect square in terms of the sum of two consecutive natural numbers. For example, 3 squared or 9 can be expressed as (4 + 5). 5 squared or 25 can be expressed as (12 + 13) and so on.

What this means, is that we can express the square of any odd number as the sum of two consecutive natural numbers. 

We cannot do this for the square of even numbers because the square of an even numbers is always an even number. An example would be 16. 16 is a perfect square and it can be expressed as (8 + 8). 8 and 8 are NOT two consecutive natural numbers but (12 + 13) are two consecutive natural numbers added together to get to 25 which is a perfect square.

One important thing to notice here is that the inverse of the rule above is not always true, so it’s not true at all. Meaning that, (12 + 13) would be 25 which is a perfect square but let’s pick to other consecutive natural number like (13 + 14) which is 27. 27 is not a perfect square. So while any odd square number can be expressed as the sum of two consecutive natural numbers, not any two consecutive natural numbers added together would result in an odd perfect square.

Expressing Square Numbers as Sum of Odd Natural Numbers

There is a relationship between the number of odd natural numbers starting from 1 added together and square numbers. That means, 1, the same things as 1 squared is the same thing as the first odd natural number. And you know that 1 is a perfect square. 4, which is the same thing as 2 squared, can be written as (1 + 3) which is the first two natural odd numbers starting from 1 added together. 9, which is the same thing as 3 squared, can be written as (1 + 3 +5), which is the same thing as the first three natural odd numbers starting from 1 added together and so on and so forth.

What this means is that if you add the first n natural odd numbers starting from 1 together, the result can be expressed as n squared.

You can use this observation in different ways in different situations to check whether a number a perfect square or not. For example, if you were to test whether 121 is a perfect square, you’d check whether 121 can be expressed as a sum of some odd natural numbers starting from 1. If you tried this, you’d find out that the number of those odd natural numbers, in this case, would be 11, namely 1 through 21 or 1, 3, 5, 7, 9, … , 21. If you add all those numbers together, you’d get 121. So you can conclude that 121 is perfect square. Moreover, sqrt(121) = 11.

The Number of Numbers from a to b and Between a and b

It’s a simple but useful skill to be able to find the number some numbers between two numbers in a number set. This could take all shapes or forms. For example, sometimes, you might need to know the number of numbers from say a to b, including a and b. I’ll talk about two of such situations here. Remember that any other type of situation that comes your way, you can use simple logic and test some simple cases and generalize your findings to more complicated cases to find the rule you’re looking for.

Calculating the Number of Numbers from a to b, Including a and b

Now, let’s say, you want to find the number of numbers from a to b, including a and b. In that case, you could think of a simple case like all the numbers from 1 to 10. Those numbers would be, 1, 2, 3, 4 5, 6, 7, 8, 9, 10. If you count them, you’ll find out that the number of those numbers is 10. So now that you know how many numbers are actually there, you could find some algebraic relationship an test it against the actual number of numbers that you have. If I subtract the first number, 1, from the last number, 10, I’d have 10 – 1 = 9 but I have 10 numbers there, not 9. So I add 1 to 9 to get to 10. Test this for any other case starting from any number and ending anywhere and it will work.

So (i) if you are working with a set of numbers like natural numbers where it’s possible to know what numbers are there and how many numbers there are between any two given numbers, and if, (ii) the numbers under study are happening based on a specific rule, then you can use the above solution to figure out the number of those numbers under study.

What (i) means is that, for example in the set of natural numbers, you know that numbers start from 1 and go all the way up to infinity. At every step, 1 is added to every number to get the next number. So the set looks something like, 1, 2, 3, … Moreover, you know that there is no number between 1 and 2 for example in this set of numbers. If you talk about the set of rational numbers, then there’s no way to know how many numbers there are between 1 and 2. The number would actually be infinite. So there, the whole solution would be useless.

What (ii) means is, that for example the numbers under study start from 1 and go all the way up to 101 or the numbers start at 1 and go all the way up to 100 missing every other number, 1, 3, 5, 7, … , 101. Here the progression of numbers is happening based on some uniform rule and so it’s possible to figure out the exact number of numbers.

Getting back to our original problem, figuring out the number of numbers from a to b, numbers happening based on a rule similar to 1, 2, 3, … , b, you could write the number of those numbers as (b – a + 1)

Calculating the Number of Numbers Between a and b, Excluding a and b

This would be the exact same situation, except that you’d need to make some changes to the formula you got in the previous step. Since in this case, you have to subtract two numbers from that sum (you have to exclude a and b from the sum), you’d end up with (b – a + 1) -2 which is equal to (b – a – 1).

 

Number of Non-square Numbers Between Two Consecutive Square Numbers

You know that square numbers are 1 squared, 2 squared, 3 squared, etc. That is, 1, 4, 9, 16, …

So now, between 1 and 4, you have two non-square numbers, namely, 2 and 3. The difference of 1 and 4 is three. So the number of non-square numbers between 1 and 4 is one less than the difference of the two numbers.

Between 4 and 9, there are four numbers, namely, 5, 6, 7 and 8. The difference of 4 and 9 is five. So again, the number of non-square numbers between 4 and 9 is one less than the difference of the two numbers.

If you do the same thing for all the square numbers, you’ll notice the same thing all the time. That is, the number of non-square numbers between two square numbers is always one less than the difference of the two numbers.

How to Calculate the Number of Non-square Numbers Between Two Square Numbers

Now if you want to calculate the number of non-square numbers between any two square numbers, you can use the following procedure:

If you take any square number as a squared, the next square number will be (a+1) squared. The difference of the two is (a+1) squared – a squared. Expand this expression and you’ll get 2a+1. One less than 2a+1 would be 2a+1-1 which is 2a.

So the number of non-squared numbers between two square numbers is always equal to 2a, provided that you take the smaller number as a. For example, if you were to calculate the number of non-square numbers between 9 squared and 10 squared, that is, between 81 and 100, you’d have to take a as 9 and not 81 and then 2a would be 2 times 9 which is 18.

 

The Relationship Between Triangular and Square Numbers

Triangular number in mathematics are the kind of numbers that would represent a right-angled triangle in the following fashion. You can click the link below and take a look at the triangle.

Click to see the Triangular Number Image

As you can see, there’s a triangle with a base and on top of the base, you have rows made of some number of marbles let’s say, and the next row, again made of marbles but as you move up in the triangle across the rows, the number of marbles is reduced by one each time you move up one row. The triangle is complete when you reach a row with just one marble.

So in this way, we can list the following triangular numbers: 1, 3, 6 , 10, 15, …

It turn out that you can add any two consecutive triangular numbers and get a square number. Let’s look at an image to see how it can be done:

Click to see how to add triangular numbers to get square numbers

So what this means is that, you can add any two consecutive triangular numbers to get a perfect square. We’ll most probably use this result later.

The Relationship Between the Number of Zeros in Numbers and their Squares

The relationship between the number of zeros in a square number and the number squared to get the square number is as follows:

If you multiply the number of zeros in a number by two, you’ll get the number of zeros there are in the square of that number.

In other words, if we take the number of zeros in a number as n, then the number of zeros in the square of that number is going to be equal to 2n.

How to Decide Whether a Number is a Perfect Square

If you look at some perfect squares, you’ll find that all those number end with one of these numbers in the 1’s digit: 0, 1, 4, 5, 6, 9.

And you’ll notice that none of those numbers ends with any of these digits in the 1’s digit: 2, 3, 7, 8

So based on this observation, you can say that,

  1. If a natural number ends with one of the digits 0, 1, 4, 5, 6 or 9, it “must” be a perfect square. Pay attention to the word must here. That means that you could find some natural number having one of those digits in the 1’s place value and if you tested that number, it might not be a perfect square.
  2. If a natural number ends with one of the digits 2, 3, 7 or 8 in the 1’s digit, it is not a perfect square.

So you can use the observation above in the following way. Meaning that you can say, if 1 is the case, then there is no way to decide whether your number is a perfect square or not whereas if 2 is the case, you can say that the number is not a perfect square.

The Relationship Between the 1’s Place in a Number and the 1’s place in the Corresponding Square Number

If you list all the square numbers between 1 and 2500 for example, you’ll notice the following pattern:

1, 9 –> 1; 2, 8 –> 4; 3, 7–>9; 4, 6–>6; 5–>5; 0–>0

That means, if the number ends in either a 1 or 9 in the 1’s place, the square of that number ends in a 1 in the 1’s place, a 2 or 8 in the 1’s place, the square of the number in a 4 in the 1’s place and so on and so forth.

Here is a summery of the relationships above:

  • If a number has a 1 or 9 in the 1’s place, the square of that number will have a 1 in the 1’s place.
  • If a number has a 4 or 6 in the 1’s place, the square of that number will have a 6 in the 1’s place.
  • And so on …