8-9 Perfect Squares

[Karri Pretty]

In mathematics, we might have come across different types of numbers such as even, odd, prime, composite, etc. However, there is a particular type of number, i.e. a perfect square. These can be identified and expressed with the help of factorisation of a number. In this article, you will learn the definition of perfect square numbers, notation, the list of these numbers between 1 and 100 and so on.

I have made the following program which asks for a user input upper bound and calculates & prints every perfect square up to the upper bound. However, I think that my is_perfect_square function is not very efficient as it takes a long time to compute the perfect squares when the upper bound is in the thousands or more. I am wondering how I can make my program more efficient, I think using the math module with to use sqrt could work but I am no mathematician so request help.My program is:

A perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer. For example, 25 is a perfect square because it is the product of integer 5 by itself, 5 5 = 25. However, 21 is not a perfect square number because it cannot be expressed as the product of two same integers.

In this article, we will discuss the concept of perfect squares and learn how to identify them. We will discuss the definition of a perfect square, its formula, and the list of perfect squares along with a few solved examples for a better understanding.

A perfect square is a positive integer that is obtained by multiplying an integer by itself. In simple words, we can say that perfect squares are numbers that are the products of integers by themselves. Generally, we can express a perfect square as x2, where x is an integer and the value of x2 is a perfect square.

Perfect squares are numbers that are obtained by squaring a whole number or an integer. Let us look at an example to understand the concept behind perfect squares. For this, we can take a set of 4 marbles and another set of 6 marbles. Let us arrange the marbles. Were you able to arrange the marbles in the way shown below?

Let's analyze this activity. We can form a square with 4 marbles such that there are 2 rows, with 2 marbles in each row. With 6 marbles, we can form a rectangle such that there are 2 rows, with 3 marbles in each row. Mathematically, it means 4 = 2 2 and 6 = 3 2. Let us focus only on the numbers which form a square. Here, 4 = 2 2 = 22. Now, if we look at the definition of a perfect square, it says, "A perfect square is a number which is obtained by squaring an integer."

The table given below shows the perfect squares of the first 20 natural numbers. The first column shows the natural number and the second column shows the square of the natural number. You can easily find the square of a natural number by multiplying it by itself. For example, 2 2 = 4, 3 3 = 9, 4 4 = 16, and so on.

Observe the last digit of the perfect square numbers 1 to 20 as given in the table above. You will notice that they end with any one of these digits 0, 1, 4, 5, 6, or 9. After trying various perfect square numbers you would have observed an important property of perfect squares. Numbers that have any of the digits 2, 3, 7, or 8 in their units place are non-perfect square numbers, whereas, numbers that have any of the digits 0, 1, 4, 5, 6, or 9 in their units place might be perfect squares. The following observations can be made to identify a perfect square.

Let us look at a few deviations from the above-defined rules of a perfect square number. The numbers 159 and 169 both end with the digit 9 but 169 is a perfect square, whereas 159 is not. If the number ends with the digit 0, then you may look for the following: How many zeros are there at the end of the number? Let's say we have a number 1000. If there is an odd number of zeros, then it's definitely not a perfect square. 1000 has 3 zeros at the end. Thus, it's not a perfect square. If there are an even number of zeros, then it might be a perfect square. 400 and 300 both have an even number of zeros at the end, but 400 = 202, which is a perfect square, but 300 is not a square of any whole number.

You can find the square of a number by multiplying it by itself, for example, 6 6 = 36, However, there are some simple methods that work for special types of numbers. These can be applied to square a number in a very short time. In other words, this can be used to calculate the square of a large number without using the long multiplication method.

Numbers Ending with Digit 5: Let's consider a number ending with 5, like 65. Now, we can find the square of 65 through a sequence of four simple steps. First, we need to separate the numbers 6 and 5. Next, multiply 6 by its successor, i.e. 7. Now for the third step, square the number 5 to get 25. Further, for the final step write the digits of the second step, followed by 25. The final answer for the square of 65 is 4225.

It is given that the number of rows is the same as the number of columns. This indicates that the chairs are arranged in the form of a square. To find the total number of chairs in the auditorium, we will find the square of 60 units. 602 = 60 60 = 3600. Therefore, there are 3600 chairs in the auditorium which is a perfect square number.

To identify which number is to be added to 75 to make it a perfect square number, we have to identify which number is a perfect square greater than 75. By looking at the perfect squares list, we know that 81 is the nearest number greater than 75 which is a perfect square. Therefore, the answer is 81 - 75 = 6.

A number is considered to be a perfect square if it can be written as a square of an integer. For example, 9 is a perfect square because 3 3 = 32 = 9. However, 21 is not a perfect square, because there is no whole number that can be squared to give 21 as the product.

A perfect square number can be factorized just as we factorize a normal number. It can be written as a product of a number by itself. For example, the number 16 can be factorized as 4 4, or it can be factorized as a product of prime numbers as 16 = 2 2 2 2.

A perfect square is a number that can be expressed as a product of a whole number by itself. The factors of 7 are 1 and 7 only. So, we cannot express 7 as a product of any integer/whole number. So, 7 is not a whole number.

A number is considered to be a perfect square if it can be written as a square of an integer. For example, 9 is a perfect square because 3 \u00d7 3 = 32 = 9. However, 21 is not a perfect square, because there is no whole number that can be squared to give 21 as the product.

A perfect square number can be factorized just as we factorize a normal number. It can be written as a product of a number by itself. For example, the number 16 can be factorized as 4 \u00d7 4, or it can be factorized as a product of prime numbers as 16 = 2 \u00d7 2 \u00d7 2 \u00d7 2.

Hi there! I am currently explaining a certain exercise to my buddy and part of it consists in creating a list of perfect squares up to root of 64 (included) therefore the list has to look like this [0, 1, 4, 9, 16, 25, 36, 49, 64]My first idea was this

Hi! thank you so much for your help my code is better now! I just do not really understand your last statement 'if you want to avoid the use of the square root, you can replace the for loop with a while loop'Isn't it the same the for loop and the wwhile loop?

One more way...change your view to Page Layout. This changes the grid scale to inches (or the system default units), and then you can specifically set both height and width to the same value (e.g. 0.25 inches). IMO page layout is the best standard view for working on the appearance of the spreadsheet.

This technique can be used to create specifically sized charts and tables for use to copy/paste into other productivity products (e.g. Word, Powerpoint, or Publisher). It provides a simple mechanism for consistent sized and ratio graphics for publication.

If you don't need to be exact, just eyeball it. If you do need to be closer than eyeballing, put a square from the Drawing toolbar on your sheet and size it. If you want it 10 x 10, use code like this:

VBA seems a little overkill for such a simple outcome..If you click and hold when you go to drag to change the row/col size, the size in pixels is shown in brackets. These units are not scaled and thus if you set the row and column sizes to equal pixels, they will be square. Of course, this is a manual process.. but you can find the equivalent sizes and then select a range of rows/columns and set all of their sizes at once.

Each red, green, and blue subpixel make up the whole pixel. As you can see, the combination of the 3 are wider than the height of 1 subpixel. In most cases, the difference is subtle, and most people might not notice it. However, in some cases, people do.

If you wanted to do it for the whole sheet, you could use this trick - which may be helpful anyway: click the box to the left of column heading A to select all cells; click on and drag one of the column header dividers to the size you want, noting the number of pixels for the resulting cell width (I'm using Excel 2007, which shows this); do the same for one of the row label dividers, matching it to the column width by pixels. This should make all cells in the sheet boxes. Which of course is not what you asked, but I had hoped this trick would work with a subset of cells. Unfortunately it doesn't.

Note that the squares disappear (by a changing column width) when you change the font type or size. This is due to the way Excel calculates the column width. See: -us/help/214123/description-of-how-column-widths-are-determined-in-excel

I wanted to make a perfect square grid for a sewing project and kept getting all kinds of weird answers for this question, so I decided to play with it myself to figure it out. I discovered it's impossible to get a perfect square, but I came as close as you can get, just a sliver off.

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