We mentioned before that when you square a number, you’ll get a square number or a perfect square. You can use the perfect square to find out what numbers were multiplied together to get the perfect square. The process of using the perfect square to get back to those numbers is finding the square root of a number. For example, 3 * 3 = 3^2 = 9. 9 Here is perfect square. You can take the square root of 9 to get back to 3. You can see that squaring a number and taking the square root of the result are actually inverse processes. Continue reading “How to Find the Square Root of a Number”
If you multiply any integer except for 0 by itself, you’ll get a positive perfect square. For example, 3 * 3 = 9 or you can write the same expression as 3^2 = 9. Now, if you have the number 9, knowing that 4 is a perfect square and want to know what number was raised to the second power or multiplied by itself to get the 9, you need a function that takes you from the perfect square back to the number. That function is called the “square root function.” That means,
3 * 3 = 9 Or 3^2 = 9, then, sqrt(9) = 3. Continue reading “Square Root Function”
Pythagorean Triplets refers to three natural numbers in such a way that the square of one of them (the greatest one) is equal to the sum of squares of the others two. For example 3, 4 and 5 form a Pythagorean triplet. 5^2 = 3^2 + 4^2. Continue reading “Pythagorean Triplets”
Calculating the square of numbers like 8, 9, etc., is easy but calculating the square of a number like 105 without using a calculator.
There is an interesting way to calculate the square of large numbers like 255 without using a calculator. It’s probably not easy to do it in your mind but you can do it on a piece of paper. It’s essentially a way to calculate the square root of numbers that have a 5 in 1’s place value. It does not matter what sort of digits you might have in the other place values but the 1’s place value has to be a 5.
So the formula will be (x5)^2 = x(x+1)100 + 25
The formula above was derived for a 2-digit number but it can be used for any number with any number of digits provided that the number has a 5 in the 1’s place. For example, if you were to calculate 115^2, you’d simply use the formula as follows:
115^2 = 11(12)100 * 25 = 13225
You can learn how this formula was created and based on that, you can create any sort of formula for any kind of situation you might find yourself in.
It turns out that you can write almost any squared number in terms of the product of two odd or even integers plus one. As an example, let’s go through the following operation:
29 * 31 = (30 – 1)(30 + 1) = 30^2 – 1 because (a – b)(a + b) = a^2 – b^2
30^2 = 29 * 31 + 1
So as you can see here, you can write 30^2 as the product of two consecutive odd natural numbers, 29 and 31 and add 1 to the product. You can do this with almost every square number. I am using the word “almost” here since I have not taken the time to test every possible case not have I proved this nor have I seen a proof for it.
Wherever this happens to come in handy depends on the kind of problems you might be solving.
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.
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.
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.
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.
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:
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 square of an odd number is always an odd number.
The square of an even number is always an even number.