Square root 5

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Square Root Calculator

Our square root calculator estimates the square root of any positive number you want. Just enter the chosen number and read the results. Everything is calculated quickly and automatically! With this tool, you can also estimate the square of the desired number (just enter the value into the second field) which may be a great help in finding perfect squares from the square root formula. Are you struggling with the basic arithmetic operations: adding square roots, subtracting square roots, multiplying square roots or dividing square roots? Not any more! In the following text, you will find a detailed explanation about different square root properties, e.g., how to simplify square roots, with many various examples given. With this article, you will learn once and for all how to find square roots!

Have you ever wondered what is the origin of the square root symbol √? We can assure you that this history is not as simple as you might think at first. The origin of the root symbol goes back to ancient times, as the origin of the percent sign.

If you're looking for the square root graph or square root function properties, head directly to the appropriate section (just click the links above!). There, we explain what is the derivative of a square root using a fundamental square root definition; we also elaborate on how to calculate square roots of exponents or square roots of fractions. Finally, if you are persistent enough, you will find out that square root of a negative number is, in fact, possible. In that way, we introduce complex numbers which find broad applications in physics and mathematics.

Square root symbol √

The operation of the square root of a number was already known in antiquity. The earliest clay tablet with the correct value of up to 5 decimal places of √2 = 1.41421 comes from Babylonia (1800 BC - 1600 BC). Many other documents show that square roots were also used by the ancient Egyptians, Indians, Greeks, and Chinese. However, the origin of the root symbol √ is still largely speculative.

  • many scholars believe that square roots originate from the letter "r" - the first letter of the Latin word radix meaning root,
  • another theory states that square root symbol was taken from the Arabic letter ج that was placed in its original form of ﺟ in the word جذر - root (the Arabic language is written from right to left).

The first use of the square root symbol √ didn't include the horizontal "bar" over the numbers inside the square root (or radical) symbol, √‾. The "bar" is known as a vinculum in Latin, meaning bond. Although the radical symbol with vinculum is now in everyday use, we usually omit this overline in the many texts, like in articles on the internet. The notation of the higher degrees of a root has been suggested by Albert Girard who placed the degree index within the opening of the radical sign, e.g., ³√ or ⁴√.

The last question is why is the square root operation called root regardless of its true origin? The explanation should become more evident if we write the equation x = ⁿ√a in a different form: xⁿ = a. x is called a root or radical because it is the hidden base of a. Thus, the word radical doesn't mean far-reaching or extreme, but instead foundational, reaching the root cause.

Square root definition

In mathematics, the traditional operations on numbers are addition, subtraction, multiplication, and division. Nonetheless, we sometimes add to this list some more advanced operations and manipulations: square roots, exponentiation, logarithmic functions and even trigonometric functions (e.g., sine and cosine). In this article, we will focus on the square root definition only.

The square root of a given number is every number whose square yields the original number . Therefore, the square root formula can be expressed as:


where is a mathematical symbol that means if and only if. Each positive real number always has two square roots - the first is positive and second is negative. However, for many practical purposes, we usually use the positive one. The only number that has one square root is zero. It is because √0 = 0 and zero is neither positive nor negative.

There is also another common notation of square roots that could be more convenient in many complex calculations. This alternative square root formula states that the square root of a number is a number raised to the exponent of the fraction one half:

In geometric interpretation, the square root of a given area of a square gives the length of its side. That's why has word square in its name. A similar situation is with the cube root . If you take the cube root of the volume of a cube, you get the length of its edges. While square roots are used when considering surface areas, cube roots are useful to determine quantities that relate to the volume, e.g., density.

How to find the square root?

Maybe we aren't being very modest, but we think that the best answer to the question how to find the square root is straightforward: use the square root calculator! You can use it both on your computer and your smartphone to quickly estimate the square root of a given number. Unfortunately, there are sometimes situations when you can rely only on yourself, what then? To prepare for this, you should remember several basic perfect square roots:

  • square root of 1: , since ;
  • square root of 4: , since ;
  • square root of 9: , since ;
  • square root of 16: , since ;
  • square root of 25: , since ;
  • square root of 36: , since ;
  • square root of 49: , since ;
  • square root of 64: , since ;
  • square root of 81: , since ;
  • square root of 100: , since ;
  • square root of 121: , since ;
  • square root of 144: , since ;

The above numbers are the simplest square roots because every time you obtain an integer. Try to remember them! But what can you do when there is a number that doesn't have such a nice square root? There are multiple solutions. First of all, you can try to predict the result by trial and error. Let's say that you want to estimate the square root of :

  1. You know that and so should be between and .
  2. Number is closer to the (effectively closer to the ) so you can try guessing that is .
  3. Then, you square obtaining (as the square root formula says) which is higher than . You have to try with a smaller number, let's say .
  4. The square of is . Now you have a smaller number, but much closer to the . If that accuracy satisfies you, you can end estimations here. Otherwise, you can repeat the procedure with a number chosen between and ,e.g., and so on and so forth.

Another approach is to simplify the square root first and then use the approximations of the prime numbers square roots (typically rounded to two decimal places):

  • square root of 2: ,
  • square root of 3: ,
  • square root of 5: ,
  • square root of 7: ,
  • square root of 11: ,
  • square root of 13: ,
  • square root of 17: ,
  • square root of 19: , etc.

Let's try and find the square root of again. You can simplify it to (you will learn how to simplify square root in the next section) and then substitute . Finally, make a multiplication . The result is the same as before!

You can check whether a number is prime or not with our prime number calculator. A prime number is a natural number (greater than one) that can't be obtained as a product of two smaller natural numbers. For example, 7 is a prime number because you can get it only by multiplying or . On the other hand, number 8 is not prime, because you can form it by multiplying or (besides product of 1 and 8 itself).

Square root calculator

In some situations, you don't need to know the exact result of the square root. If this is the case, our square root calculator is the best option to estimate the value of every square root you desired. For example, let's say you want to know whether is greater than . From the calculator, you know that , so . It is very close to the , but it isn't greater than it! The square root calculator gives the final value with relatively high accuracy (to five digits in above example). With the significant figure calculator, you can calculate this result to as many significant figures as you want.

Remember that our calculator automatically recalculates numbers entered into either of the fields. You can find what is the square root of a specific number by filling the first window or get the square of a number that you entered in the second window. The second option is handy in finding perfect squares that are essential in many aspects of math and science. For example, if you enter in the second field, you will find out that is a perfect square.

In some applications of the square root, particularly those pertaining to sciences such as chemistry and physics, the results are preferred in scientific notation. In brief, an answer in scientific notation must have a decimal point between the first two non-zero numbers and will be represented as the decimal multiplied by 10 raised to an exponent. For example, the number is written as in scientific notation, whereas is written as in scientific notation. The results obtained using the square root calculator can be converted to scientific notation with the scientific notation calculator.

How to simplify square roots?

First, let's ask ourselves which square roots can be simplified. To answer it, you need to take the number which is after the square root symbol and find its factors. If any of its factors are square numbers (4, 9, 16, 25, 36, 49, 64 and so on), then you can simplify the square root. Why are these numbers square? They can be respectively expressed as 2², 3², 4², 5², 6², 7² and so on. According to the square root definition, you can call them perfect squares. We've got a special tool called the factor calculator which might be very handy here. Let's take a look at some examples:

  • can you simplify √27? With the calculator mentioned above, you obtain factors of 27: 1, 3, 9, 27. There is 9 here! This means you can simplify √27.
  • can you simplify √15? Factors of 15 are 1, 3, 5, 15. There are no perfect squares in those numbers, so this square root can't be simplified.

So, how to simplify square roots? To explain that, we will use a handy square root property we have talked about earlier, namely, the alternative square root formula:

We can use those two forms of square roots and switch between them whenever we want. Particularly, we remember that power of multiplication of two specific numbers is equivalent to the multiplication of those specific numbers raised to the same powers. Therefore, we can write:


How can you use this knowledge? The argument of a square root is usually not a perfect square you can easily calculate, but it may contain a perfect square amongst its factors. In other words, you can write it as a multiplication of two numbers, where one of the numbers is the perfect square, e.g., (9 is a perfect square). The requirement of having at least one factor that is a perfect square is necessary to simplify the square root. At this point, you should probably know what the next step will be. You need to put this multiplication under the square root. In our example:


You have successfully simplified your first square root! Of course, you don't have to write down all these calculations. As long as you remember that square root is equivalent to the power of one half, you can shorten them. Let's practice simplifying square roots with some other examples:

  • How to simplify square root of 27? ;
  • How to simplify square root of 8? ;
  • How to simplify square root of 144? .

In the last example, you didn't have to simplify the square root at all, because 144 is a perfect square. You could just remember that 12 * 12 = 144. However, we wanted to show you that with the process of simplification, you can easily calculate square roots of perfect squares too. It is useful when dealing with big numbers.

Finally, you may ask how to simplify roots of higher orders, e.g., cube roots. In fact, the process is very analogical to the square roots, but in the case of cube roots, you have to find at least one factor that is a perfect cube, not a perfect square, i.e., 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³ and so on. Then you divide your number into two parts and put under the cube root. Let's take the following example of simplifying ³√192:

It may seem a little bit complicated at first glance, but after some practice, you will be able to simplify roots in your head. Trust us!

Adding, subtracting, multiplying and dividing square roots

Adding square roots and subtracting square roots

Unfortunately, adding or subtracting square roots are not as easy as adding/subtracting regular numbers. For example, if 2 + 3 = 5, it doesn't mean that √2 + √3 equals √5. That's wrong! To understand why is that, imagine that you have two different types of shapes: triangles 🔺 and circles 🔵. What happens when you add one triangle to one circle 🔺 + 🔵? Nothing! You still have one triangle and one circle 🔺 + 🔵. On the other hand, what happens when you try to add three triangles to five triangles: 3🔺 + 5🔺? You'll we get eight triangles 8🔺.

Adding square roots is very similar to this. The result of adding √2 + √3 is still √2 + √3. You can't simplify it further. It is a different situation however when both square roots have the same number under the root symbol. Then we can add them just as regular numbers (or triangles). For example 3√2 + 5√2 equals 8√2. The same thing is true subtracting square roots. Let's take a look at more examples illustrating this square root property:

  • What is ? Answer: ;
  • What is ? Answer: ;
  • What is ? Answer: , because we simplified √8 = √(4 * 2) = √4 * √2 = 2√2;
  • What is ? Answer: , because we simplified √45 = √(9 * 5) = √9 * √5 = 3√5 and √20 = √(4 * 5) = √4 * √5 = 2√5;
  • What is ? Answer: , we can't simplify this further;
  • What is ? Answer: , we can't simplify this further than this, but we at least simplified √18 = √(9 * 2) = √9 * √2 = 3√2.

Multiplying square roots and dividing square roots

Now, when adding square roots is a piece of cake for you, let's go one step further. What about multiplying square roots and dividing square roots? Don't be scared! In fact, you already did it during the lesson of simplifying square roots. Multiplying square roots is based on the square root property that we have used before a few times, that is:

Do you remember how to multiply numbers that are raised to the same power? As a reminder:


and therefore


As opposed to addition, you can multiply every two square roots. Remember that multiplication has commutative properties, that means that the order to which two numbers are multiplied does not matter. Few examples should clarify this issue:

  • What is ? Answer: ;
  • What is ? Answer: , because multiplication is commutative;
  • What is ? Answer: , we simplified √18 = √(9 * 2) = √9 * √2 = 3√2.

Dividing square root is almost the same since:


All you need to do is to replace multiplication sign with a division. However, the division is not a commutative operator! You have to calculate the numbers that stand before the square roots and numbers under the square roots separately. As always, some practical examples:

  • What is ? Answer: ;
  • What is ? Answer: ;
  • What is ? Answer: , we switched there from a simple fraction 2/5 to the decimal fraction 2/5 = 4/10 = 0.4.

Square roots of exponents and fractions

Calculating the square root of the exponent or square root of the fraction might not be clear for you. But with the knowledge you acquired in the previous section, you should find it easier than you expected! Let's begin with the square roots of exponents. In that case, it will be easier for you to use the alternative form of square root . Do you remember the power rule? If not, here is a quick reminder:


where and are any real numbers. Now, when you place instead of you'll get nothing else but a square root:


and that's how you find the square root of an exponent. Speaking of exponents, above equation looks very similar to the standard normal distribution density function, which is widely used in statistics.

If you're still not sure about taking square roots of exponents, here are a few examples:

  • square root of 2^4: ,
  • square root of 5^3: ,
  • square root of 4^5: .

As you can see, sometimes it is impossible to get a pretty result like the first example. However, in the third example, we showed you a little trick with expressing as . This approach can often simplify more complicated equations.

What about square roots of fractions? Take a look at the previous section where we wrote about dividing square roots. You can find there the following relation that should explain everything:


where is a fraction. Below you can find some examples of square roots of a fraction:

  • square root of 4/9: ,
  • square root of 1/100: ,
  • square root of 1/5: .

Leaving roots in the denominator is not a very good habit. That's why we got rid of it in the last example. We just multiplied both the numerator and denominator by the same number (we can always do that, as the number we multiply by equals 1), in this case by .

Square root function and graph

Functions play a vital role not only in mathematics but in many other areas like physics, statistics, or finance. Function is nothing more than a formula that says how the value of changes with the argument . To see some examples, check out our finance tools made by financial specialists, for example, the compound interest calculator or future value calculator. You will find there some functions that you can apply in real life. They're a great help if you want to know how to calculate the compound interest or to estimate the future value of an annuity.

Below you can find the square root graph, made up of half of a parabola. Check it and try to validate, for example, whether the square root function of is and of is (as it should be).

Let's go back to the square root function and explore what are its basic properties. We consider there only the positive part of (as you can see in the square root graph above). So, the square root function:

  • is continuous and growing for all non-negative ,
  • is differentiable for all positive (see the derivative of the square root section for more information),
  • approaches the limit of infinity as approaches infinity ( when ),
  • is a real number for all non-negative and a complex number for all negative (we write more about it in the square root of a negative number section).

You probably have already noticed that the square root of the area of a square gives its side length. This feature is used in one of our construction calculators - square footage calculator. If you plan to do any renovation in the future, these tools might be a great help. Don't forget to use them!

Derivative of the square root

A derivative of a function tells us how fast this function changes with its argument. One of the simplest examples in physics is the position of an object and its velocity (the rate of change of position). Let's say that the function describes how the distance of the moving car from a specific point changes with time . Do you know what determines how fast the change is in your distance traveled? The answer is the speed of the car! So the derivative of the position is velocity (velocity can depend on time too). To denote derivative, we usually use apostrophe or the derivative symbol .

The derivative of the general function is not always easy to calculate. However, in some circumstances, if the function takes a specific form, we've got some formulas. For example, if


where is any real number, the derivative is as follows:


It may not look like, but this answers the question what is the derivative of a square root. Do you remember the alternative (exponential) form of a square root? Let us remind you:


You can see that in this case , so the derivative of a square root is:


Since a number to a negative power is one over that number, the estimation of the derivation will involve fractions. We've got a tool that could be essential when adding or subtracting fractions with different denominators. It is called the LCM calculator, and it tells you how to find the Least Common Multiple.

The derivative of a square root is needed to obtain the coefficients in the so-called Taylor expansion. We don't want to dive into details too deeply, so, briefly, the Taylor series allows you to approximate various functions with the polynomials that are much easier to calculate. For example, the Taylor expansion of about the point is given by:


which is valid for . Although the above expression has an infinite number of terms, to get the approximate value you can use just a few first terms. Let's try it! With and first five terms, you get:



and the real value, provided by our calculator, is . Close enough!

That was a lot of maths and equations so far. For those of you who are persistent enough, we've prepared the next section which explains how to calculate the square root of a negative number.

Square root of a negative number

At school, you probably have been taught that square root of a negative number does not exist. This is true when you consider only real numbers. A long time ago, to perform advanced calculations, mathematicians had to introduce a more general set of numbers - the complex numbers. They can be expressed in the following form:


where is the complex number with the real part and imaginary part . What differs between a complex number and a real one is the imaginary number . Here you have some examples of complex numbers: , , , . You may be surprised seeing there which is a real number. Yes, it is, but it is also a complex number with . Complex numbers are a generalization of the real numbers.

So far imaginary number is probably still a mystery for you. What is it at all? Well, although it may look weird, it is defined by the following equation:


and that's all that you need to calculate the square root of every number, whether it is positive or not. Let's see some examples:

  • square root of -9: ,
  • square root of -13: ,
  • square root of -49: .

Isn't that simple? This problem doesn't arise with the cube root since you can obtain the negative number by multiplying three of the identical negative numbers (which you can't do with two negative numbers). For example:


That's probably everything you should know about square roots. We appreciate that you stayed with us until this point! As a reward you should bake something sweet for yourself :-) Check out our perfect pancake calculator to find out how to make the perfect pancake, however you like it. You may need our grams to cups calculator to help you with this. It works both ways, i.e., to convert grams to cups and convert cups to grams. And if you ask yourself "How many calories should I eat a day?", visit our handy calorie calculator!


Can a number have more than one square root?

Yes, in fact all positive numbers have 2 square roots, one that is positive and another that is equal but negative to the first. This is because if you multiply two negatives together, the negatives cancel and the result is positive.

How do you find the square root without a calculator?

  1. Make an estimate of the square root. The closest square number is acceptable if you’re at a loss.
  2. Divide the number you want to find the square root of by the estimate.
  3. Add the estimate to the result of step 2.
  4. Divide the result of step 3 by 2. This is your new estimate.
  5. Repeat steps 2-4 with you new estimate. The more times this is repeated, the more accurate the result is.

How can I estimate square roots?

  1. Find the nearest square number above and below the number you are thinking of.
  2. The square root will be between the square roots of these numbers.
  3. The closeness of the number to a square root indicates how close the root is. E.g., 26 is very close to 25, so the root will be very close to 5.
  4. Try a few times to get the hang of it.

Is the square root of 2 a rational number?

No, the square root of 2 is not rational. This is because when 2 is written as a fraction, 2/1, it can never have only even exponents, and therefore a rational number cannot have been squared to create it.

How can I get rid of a square root?

In algebra, squaring both sides of the equation will get rid of any square roots. The result of this operation is that the square roots will be replaced with whatever number they were finding the square root of.

Are square roots rational?

Some square roots are rational, whereas others are not. You can work out if a square root is rational or not by finding out if the number you are square rooting can be expressed in terms of only even exponents (e.g. 4 = 22 / 1 2). If it can, its root is rational.

Is the square root of 5 a rational number?

The square root of 5 is not a rational number. This is because 5 cannot be expressed as a fraction where both the numerator and denominator have even exponents. This means that a rational number cannot have been squared to get 5.

Is the square root of 7 a rational number?

The result of square rooting 7 is an irrational number. 7 cannot be written as a fraction with only even exponents, meaning that the number squared to reach 7 cannot be expressed as a fraction of integers, and therefore is not rational.

What is the derivative of the square root of x?

The derivative of square root x is x-1/2/2, or 1/2SQRT(x). This is because the square root of x can be expressed as x1/2, from which differentiation occurs normally.

How do you find the square root of a decimal?

  1. Convert the decimal into a fraction.
  2. Find any square roots of the fraction, or estimate it. Make the fraction equal to the square root you found squared.
  3. Cancel the square root and the square leaving you with the fraction.
  4. Rewrite the fraction as a decimal as your final answer.
Sours: https://www.omnicalculator.com/math/square-root

How to Find the Square Root of 5?

This question might be bothering you for quite some time now. The simplest way to find the square root of any number would be by using the division method. How to find the value of root 5? Follow the steps given below:

Step 1: The first step is to group the digits in pairs of two. You start from the unit that is in the unit place and move towards the left-hand side for a number before the decimal point. For the number after the decimal point, you group the first two numbers and move towards the right-hand side.

5. 00 00 00 00 

Step 2: In this step, you will have to pick the largest square number that is either equal to or lesser than the first number pair. Now take this number as the divisor and also note down the quotient. 

Step 3: Now, you subtract the final product of the quotient and the divisor and the quotient from the pair of numbers or the number. Next, you bring down the next pair of numbers.

Step 4: You now need to calculate the divisor. To do that, you’ll have to multiply the previous quotient by 2 and then pick a new number in such a way that the digit and the new divisor is less than or equal to the new dividend

Step 5: Repeat Step 2, Step 3, and Step 4, until all the pairs of numbers are exhausted. Now, the quotient that you’ve found is the square root. In case of the value of under root 5, this is how it is done. 




5 . 00 00 00 



+  2

1 00

- 84


+    3

  16 00

-  13 29


+    6


-     26796



Therefore, the square root of 5 = 2.236

What is the Square Root of 5?

The value of root 5, when reduced to 5 decimal points, is 2.23606 and this is just the simplified version of the value. In addition to that, the actual value of root 5 can be equal to at least ten billion digits. 

Sample Questions

1. Using the division method, find the square root of the value 784.









- 384



2. Using the division method, find the square root of the value 5329





53 29



  4 29

-  4 29


3. Find the square root of 66049





6 60 49



+  5

2 60

2 25


  35 49

-  35 49


Frequently Asked Questions

1. How to Find the Value of Root 5?

Solution: Follow the steps given below:

Step 1: The first step is to group the digits in pairs of two. You start from the unit that is in the unit place and move towards the left-hand side for a number before the decimal point. For the number after the decimal point, you group the first two numbers and move towards the right-hand side.

Step 2: In this step, you will have to pick the largest square number that is either equal to or lesser than the first number pair. Now take this number as the divisor and also note down the quotient. 

Step 3: Now, you subtract the final product of the quotient and the divisor and the quotient from the pair of numbers or the number. Next, you bring down the next pair of numbers.

Step 4: You now need to calculate the divisor. To do that, you’ll have to multiply the previous quotient by 2 and then pick a new number in such a way that the digit and the new divisor is less than or equal to the new dividend

Step 5: Repeat Step 2, Step 3, and Step 4, until all the pairs of numbers are exhausted. Now, the quotient that you’ve found is the square root. In case of the value of under root 5, this is how it is done. 




5 . 00 00 00 



+  2

1 00

- 84


+    3

  16 00

-  13 29


+    6


-     26796



Therefore, the square root of 5 = 2.236

2. What is the Square Root of 5?

Solution : The value of root 5, when reduced to 5 decimal points, is 2.23606 and this is just the simplified version of the value.

Sours: https://www.vedantu.com/maths/square-root-of-5
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Square Root of 5

The square root of 5 is expressed as √5 in the radical form and as (5)½ or (5)0.5 in the exponent form. The square root of 5 rounded up to 5 decimal places is 2.23607. It is the positive solution of the equation x2 = 5.

  • Square Root of 5: 2.23606797749979
  • Square Root of 5 in exponential form: (5)½ or (5)0.5
  • Square Root of 5 in radical form: √5

What Is the Square Root of 5?

Let us first understand the meaning of square root. The square root of a number is the number which, when multiplied to itself, gives the product as the original number. Consider the example:

Here 5 is called the square root of 25. 25 is a perfect square. So the square root of 25 is 5. Now, what is the square root of 5? Does that mean non-square numbers cannot have a square root? Non-square numbers also have a square root, just that they are not whole numbers. For real numbers a and b,

The square root of 5 in the radical form is expressed as √5 and in exponent form, it is expressed as 5½. The square root of 25 is the inverse operation of squaring 5 and -5

  • 5 × 5=25
  • (-5) × (-5) = 25. 

Let us look at the square root of 5

Square Root of 5

We know that factors of 5 are 5 × 1 = 5

  • √5 = 2.23
  • 5 is not a perfect square.

Is the Square Root of 5 Rational or Irrational?

A number that can be expressed as a ratio of two integers, that is, p/q, q ≠ 0 is called a rational number. Now let us look at the square root of 25. √25 = 5 = 5/1. Thus, √25 is a rational number. Now let us look at the square root of 5

A number that cannot be expressed as a ratio of two integers is called an irrational number.

  • 5 is not a perfect square.
  • The square root of 5 is an irrational number.

How to Find the Square Root of 5?

There are different methods to find the square root of 5. The first method is by prime factorization and the second is the conventional long division method.

Square Root of 5 Using Prime Factorization

Let us find the square root of 5 using prime factorization:

Taking square root

Let us now try finding the square root of 5 by the long division method.

Square Root of 5 By Long Division

Let us follow these steps to find the square root of 5 by the long division method.

  • Step 1: Group the digits into pairs (for digits to the left of the decimal point, pair them from right to left) by placing a bar over them. Since our number is 5, let us represent it inside the division symbol.
  • Step 2: Find the largest number such that when you multiply it with itself, the product is less than or equal to 5. We know that  2 × 2 is 4 and is less than 5. Now let us divide 5 by 2
  • Step 3: Let us place a decimal point and pairs of zeros and continue our division. Now, multiply the quotient by 2 and the product becomes the starting digit of our next divisor.
  • Step 4: Choose a number in the unit's place for the new divisor such that its product with a number is less than or equal to 100. We know that 2 is in the ten's place and our product has to be 100 and the closest multiplication is 42 × 2 = 84
  • Step 5:  Bring down the next pair of zeros and multiply the quotient 22 (ignore the decimal) by 2, which is 44 and the starting digit of the new divisor.
  • Step 6: Choose the largest digit in the unit's place for the new divisor such that the product of the new divisor with the digit at one's place is less than or equal to 1600. We see that 443, when multiplied by 3, gives 1329 which is less than 1600. Our long division now looks like

square root of 5 using long division method stepwise

  • Step 7: Add more pairs of zeros and repeat the process of finding the new divisor and product as in step 2

Note that the square root of 5 is an irrational number, i.e, it is never-ending. So, stop the process after 4 or 5 iterations, and you have the square root of 5 by the long division method.

Explore Square roots using illustrations and interactive examples

  • Evaluate the following:

    a) 5√25 + 5√4 + 5√16
    b) 5√5 + 7√5 - 10√5
    c) 5√6 + 5√25 - 5

  • The square root of 5 in the radical form is expressed as √5.
  • In exponent form, the square root of 5 is expressed as 5½.

Square Root of 5 Solved Examples

  1. Example 1: The area of a square shaped bed is 25m2. Help Alex find the side length of the bed.


    Area of a square = side × side = side2
    √25 = Side = 5
    The side length of the bed is 5m.

  2. Example 2: How will Joe prove that square root of 5 is an irrational number and square root of 25 is a rational number?


    The square root of 5 on long division gives value, √5 = 2.23(approximately). While on the other hand if Joe finds square root of 25 using long division, he will follow the below given process:

    • Make a pair of digits of 25 starting with a digit at one's place. Put a bar on each pair. Now we have to multiply a number by itself such that the product is less than or equal to 25. Here, 5 × 5 = 25 ≤ 25 so the divisor is 5 and quotient is 5. Now do the division and get the remainder as 0.
    • Square root of 25 is 5 (√25 = 5).

    square root of 25 by long division method

  3. Example 3

    If the side of a square shape wall clock is 2.33m. What is the area of a square shaps wall clock? Round the answer to the nearest whole number.


    One side of square shape wall clock = 2.23m
    Area of square = side2
    Area of a square shape wall clock = (2.23)2 = 4.9729 = 5m2

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FAQs on the Square Root of 5

What is the Value of the Square Root of 5?

The square root of 5 is 2.23606.

Why is the Square Root of 5 an Irrational Number?

The number 5 is prime. This implies that the number 5 is pairless and is not in the power of 2. Therefore, the square root of 5 is irrational.

What is the Square Root of -5?

The square root of -5 is an imaginary number. It can be written as √-5 = √-1 × √5 = i √5 = 2.236i
where i = √-1 and it is called the imaginary unit.

What is the Square Root of 5 in Simplest Radical Form?

The number 5 is a prime number. This implies that the number 5 is pairless and is not in the power of 2. Therefore, the radical form of square root of 5 cannot be simplified further.

What is the Value of 3 square root 5?

The square root of 5 is 2.236. Therefore, 3 √5 = 3 × 2.236 = 6.708.

Evaluate 12 plus 15 square root 5

The given expression is 12 + 15 √5. We know that the square root of 5 is 2.236. Therefore, 12 + 15 √5 = 12 + 15 × 2.236 = 12 + 33.541 = 45.541

Sours: https://www.cuemath.com/algebra/square-root-of-5/

Square root of 5

Positive real number which when multiplied by itself gives 5

Algebraic form{\sqrt {5}}
Continued fraction2 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \ddots}}}}

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

\sqrt{5}. \,

It is an irrationalalgebraic number.[1] The first sixty significant digits of its decimal expansion are:

2.23606797749978969640917366873127623544061835961152572427089… (sequence A002163 in the OEIS).

which can be rounded down to 2.236 to within 99.99% accuracy. The approximation 161/72 (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than 1/10,000 (approx. 4.3×10−5). As of November 2019, its numerical value in decimal has been computed to at least 2,000,000,000,000 digits.[2]

Proofs of irrationality[edit]

1. This irrationality proof for the square root of 5 uses Fermat's method of infinite descent:

Suppose that √5 is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as m/n for natural numbers m and n. Then √5 can be expressed in lower terms as 5n − 2m/m − 2n, which is a contradiction.[3] (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives 5n2 = m2 and m/n = √5, which is true by the premise. The second fractional expression for √5 is in lower terms since, comparing denominators, m − 2n < n since m < 3n since m/n < 3 since √5 < 3. And both the numerator and the denominator of the second fractional expression are positive since 2 < √5 < 5/2 and m/n = √5.)

2. This irrationality proof is also a proof by contradiction:

Suppose that √5 = a/b where a/b is in reduced form.
Thus 5 = a2/b2 and 5b2 = a2. If b were even, b2, a2, and a would be even making the fraction a/bnot in reduced form. Thus b is odd, and by following a similar process, a is odd.
Now, let a = 2m + 1 and b = 2n + 1 where m and n are integers.
Substituting into 5b2 = a2 we get:
{\displaystyle 5(2n+1)^{2}=(2m+1)^{2}}
which simplifies to:
{\displaystyle 5\left(4n^{2}+4n+1\right)=4m^{2}+4m+1}
{\displaystyle 20n^{2}+20n+5=4m^{2}+4m+1}
By subtracting 1 from both sides, we get:
{\displaystyle 20n^{2}+20n+4=4m^{2}+4m}
which reduces to:
{\displaystyle 5n^{2}+5n+1=m^{2}+m}
In other words:
{\displaystyle 5n(n+1)+1=m(m+1)}
The expression x(x + 1) is even for any integer x (since either x or x + 1 is even). So this says that 5 × even + 1 = even, or odd = even. Since there is no integer that is both even and odd, we have reached a contradiction and √5 is thus irrational.

Continued fraction[edit]

It can be expressed as the continued fraction

{\displaystyle [2;4,4,4,4,4,\ldots ]=2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+\dots }}}}}}}}.} (sequence A040002 in the OEIS)

The convergents and semiconvergents of this continued fraction are as follows (the black terms are the semiconvergents):

{\displaystyle {\color {red}{\frac {2}{1}}},{\frac {7}{3}},{\color {red}{\frac {9}{4}}},{\frac {20}{9}},{\frac {29}{13}},{\color {red}{\frac {38}{17}}},{\frac {123}{55}},{\color {red}{\frac {161}{72}}},{\frac {360}{161}},{\frac {521}{233}},{\color {red}{\frac {682}{305}}},{\frac {2207}{987}},{\color {red}{\frac {2889}{1292}}},\dots }

Convergents of the continued fraction are colored red; their numerators are 2, 9, 38, 161, ... (sequence A001077 in the OEIS), and their denominators are 1, 4, 17, 72, ... (sequence A001076 in the OEIS).

Each of these is the best rational approximation of √5; in other words, it is closer to √5 than any rational with a smaller denominator.

Babylonian method[edit]

When √5 is computed with the Babylonian method, starting with r0 = 2 and using rn+1 = 1/2(rn + 5/rn), the nth approximant rn is equal to the 2nth convergent of the convergent sequence:

\frac{2}{1} = 2.0,\quad \frac{9}{4} = 2.25,\quad \frac{161}{72} = 2.23611\dots,\quad \frac{51841}{23184} = 2.2360679779 \ldots

Nested square expansions[edit]

The following nested square expressions converge to {\sqrt {5}}:

{\displaystyle {\begin{aligned}{\sqrt {5}}&=3-10\left({\frac {1}{5}}+\left({\frac {1}{5}}+\left({\frac {1}{5}}+\left({\frac {1}{5}}+\cdots \right)^{2}\right)^{2}\right)^{2}\right)^{2}\\&={\frac {9}{4}}-4\left({\frac {1}{16}}-\left({\frac {1}{16}}-\left({\frac {1}{16}}-\left({\frac {1}{16}}-\cdots \right)^{2}\right)^{2}\right)^{2}\right)^{2}\\&={\frac {9}{4}}-5\left({\frac {1}{20}}+\left({\frac {1}{20}}+\left({\frac {1}{20}}+\left({\frac {1}{20}}+\cdots \right)^{2}\right)^{2}\right)^{2}\right)^{2}\end{aligned}}}

Relation to the golden ratio and Fibonacci numbers[edit]

The √5/2diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.

The golden ratioφ is the arithmetic mean of 1 and √5.[4] The algebraic relationship between √5, the golden ratio and the conjugate of the golden ratio (Φ = –1/φ = 1 − φ) is expressed in the following formulae:

{\displaystyle {\begin{aligned}{\sqrt {5}}&=\varphi -\Phi =2\varphi -1=1-2\Phi \\[5pt]\varphi &={\frac {1+{\sqrt {5}}}{2}}\\[5pt]\Phi &={\frac {1-{\sqrt {5}}}{2}}.\end{aligned}}}

(See the section below for their geometrical interpretation as decompositions of a √5 rectangle.)

√5 then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio:

{\displaystyle F(n)={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}.}

The quotient of √5 and φ (or the product of √5 and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the Lucas numbers:[5]

{\displaystyle {\begin{aligned}{\frac {\sqrt {5}}{\varphi }}=\Phi \cdot {\sqrt {5}}={\frac {5-{\sqrt {5}}}{2}}&=1.3819660112501051518\dots \\&=[1;2,1,1,1,1,1,1,1,\ldots ]\\[5pt]{\frac {\varphi }{\sqrt {5}}}={\frac {1}{\Phi \cdot {\sqrt {5}}}}={\frac {5+{\sqrt {5}}}{10}}&=0.72360679774997896964\ldots \\&=[0;1,2,1,1,1,1,1,1,\ldots ].\end{aligned}}}

The series of convergents to these values feature the series of Fibonacci numbers and the series of Lucas numbers as numerators and denominators, and vice versa, respectively:

{\displaystyle {\begin{aligned}&{1,{\frac {3}{2}},{\frac {4}{3}},{\frac {7}{5}},{\frac {11}{8}},{\frac {18}{13}},{\frac {29}{21}},{\frac {47}{34}},{\frac {76}{55}},{\frac {123}{89}}},\ldots \ldots [1;2,1,1,1,1,1,1,1,\ldots ]\\[8pt]&{1,{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{7}},{\frac {8}{11}},{\frac {13}{18}},{\frac {21}{29}},{\frac {34}{47}},{\frac {55}{76}},{\frac {89}{123}}},\dots \dots [0;1,2,1,1,1,1,1,1,\dots ].\end{aligned}}}


Geometrically, √5 corresponds to the diagonal of a rectangle whose sides are of length 1 and 2, as is evident from the Pythagorean theorem. Such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. Together with the algebraic relationship between √5 and φ, this forms the basis for the geometrical construction of a golden rectangle from a square, and for the construction of a regular pentagon given its side (since the side-to-diagonal ratio in a regular pentagon is φ).

Forming a dihedralright angle with the two equal squares that halve a 1:2 rectangle, it can be seen that √5 corresponds also to the ratio between the length of a cubeedge and the shortest distance from one of its vertices to the opposite one, when traversing the cube surface (the shortest distance when traversing through the inside of the cube corresponds to the length of the cube diagonal, which is the square root of three times the edge).[citation needed]

The number √5 can be algebraically and geometrically related to √2 and √3, as it is the length of the hypotenuse of a right triangle with catheti measuring √2 and √3 (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the centre point of a cube, one of its vertices, and the middle point of a side located on one the faces containing that vertex and opposite to it, are in the ratio √2:√3:√5. This follows from the geometrical relationships between a cube and the quantities √2 (edge-to-face-diagonal ratio, or distance between opposite edges), √3 (edge-to-cube-diagonal ratio) and √5 (the relationship just mentioned above).

A rectangle with side proportions 1:√5 is called a root-five rectangle and is part of the series of root rectangles, a subset of dynamic rectangles, which are based on √1 (= 1), √2, √3, √4 (= 2), √5… and successively constructed using the diagonal of the previous root rectangle, starting from a square.[6] A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ).[7] It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between √5, φ and Φ mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length √5/2 to both sides.


Like √2 and √3, the square root of 5 appears extensively in the formulae for exact trigonometric constants, including in the sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.[8] The simplest of these are

{\displaystyle {\begin{aligned}\sin {\frac {\pi }{10}}=\sin 18^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}-1)={\frac {1}{{\sqrt {5}}+1}},\\[5pt]\sin {\frac {\pi }{5}}=\sin 36^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5-{\sqrt {5}})}},\\[5pt]\sin {\frac {3\pi }{10}}=\sin 54^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}+1)={\frac {1}{{\sqrt {5}}-1}},\\[5pt]\sin {\frac {2\pi }{5}}=\sin 72^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5+{\sqrt {5}})}}\,.\end{aligned}}}

As such the computation of its value is important for generating trigonometric tables.[citation needed] Since √5 is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a dodecahedron.[citation needed]

Diophantine approximations[edit]

Hurwitz's theorem in Diophantine approximations states that every irrational numberx can be approximated by infinitely many rational numbersm/n in lowest terms in such a way that

 \left|x - \frac{m}{n}\right| < \frac{1}{\sqrt{5}\,n^2}

and that √5 is best possible, in the sense that for any larger constant than √5, there are some irrational numbers x for which only finitely many such approximations exist.[9]

Closely related to this is the theorem[10] that of any three consecutive convergentspi/qi, pi+1/qi+1, pi+2/qi+2, of a number α, at least one of the three inequalities holds:

{\displaystyle \left|\alpha -{p_{i} \over q_{i}}\right|<{1 \over {\sqrt {5}}q_{i}^{2}},\qquad \left|\alpha -{p_{i+1} \over q_{i+1}}\right|<{1 \over {\sqrt {5}}q_{i+1}^{2}},\qquad \left|\alpha -{p_{i+2} \over q_{i+2}}\right|<{1 \over {\sqrt {5}}q_{i+2}^{2}}.}

And the √5 in the denominator is the best bound possible since the convergents of the golden ratio make the difference on the left-hand side arbitrarily close to the value on the right-hand side. In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.[10]


The ringℤ[√−5] contains numbers of the form a + b√−5, where a and b are integers and √−5 is the imaginary numberi√5. This ring is a frequently cited example of an integral domain that is not a unique factorization domain.[citation needed] The number 6 has two inequivalent factorizations within this ring:

6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}). \,

The fieldℚ[√−5], like any other quadratic field, is an abelian extension of the rational numbers. The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination of roots of unity:

{\displaystyle {\sqrt {5}}=e^{{\frac {2\pi }{5}}i}-e^{{\frac {4\pi }{5}}i}-e^{{\frac {6\pi }{5}}i}+e^{{\frac {8\pi }{5}}i}.\,}

Identities of Ramanujan[edit]

The square root of 5 appears in various identities discovered by Srinivasa Ramanujan involving continued fractions.[11][12]

For example, this case of the Rogers–Ramanujan continued fraction:

{\displaystyle {\cfrac {1}{1+{\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-4\pi }}{1+{\cfrac {e^{-6\pi }}{1+\ddots }}}}}}}}=\left({\sqrt {\frac {5+{\sqrt {5}}}{2}}}-{\frac {{\sqrt {5}}+1}{2}}\right)e^{\frac {2\pi }{5}}=e^{\frac {2\pi }{5}}\left({\sqrt {\varphi {\sqrt {5}}}}-\varphi \right).}
{\displaystyle {\cfrac {1}{1+{\cfrac {e^{-2\pi {\sqrt {5}}}}{1+{\cfrac {e^{-4\pi {\sqrt {5}}}}{1+{\cfrac {e^{-6\pi {\sqrt {5}}}}{1+\ddots }}}}}}}}=\left({{\sqrt {5}} \over 1+{\sqrt[{5}]{5^{\frac {3}{4}}(\varphi -1)^{\frac {5}{2}}-1}}}-\varphi \right)e^{\frac {2\pi }{\sqrt {5}}}.}
{\displaystyle 4\int _{0}^{\infty }{\frac {xe^{-x{\sqrt {5}}}}{\cosh x}}\,dx={\cfrac {1}{1+{\cfrac {1^{2}}{1+{\cfrac {1^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {3^{2}}{1+{\cfrac {3^{2}}{1+\ddots }}}}}}}}}}}}}}.}

See also[edit]


  1. ^Dauben, Joseph W. (June 1983) Scientific AmericanGeorg Cantor and the origins of transfinite set theory. Volume 248; Page 122.
  2. ^Yee, Alexander. "Records Set by y-cruncher".
  3. ^Grant, Mike, and Perella, Malcolm, "Descending to the irrational", Mathematical Gazette 83, July 1999, pp.263-267.
  4. ^Browne, Malcolm W. (July 30, 1985) New York TimesPuzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1. (Note: this is a widely cited article).
  5. ^Richard K. Guy: "The Strong Law of Small Numbers". American Mathematical Monthly, vol. 95, 1988, pp. 675–712
  6. ^Kimberly Elam (2001), Geometry of Design: Studies in Proportion and Composition, New York: Princeton Architectural Press, ISBN 
  7. ^Jay Hambidge (1967), The Elements of Dynamic Symmetry, Courier Dover Publications, ISBN 
  8. ^Julian D. A. Wiseman, "Sin and cos in surds"
  9. ^LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR 0080682
  10. ^ abKhinchin, Aleksandr Yakovlevich (1964), Continued Fractions, University of Chicago Press, Chicago and London
  11. ^Ramanathan, K. G. (1984), "On the Rogers-Ramanujan continued fraction", Indian Academy of Sciences. Proceedings. Mathematical Sciences, 93 (2): 67–77, doi:10.1007/BF02840651, ISSN 0253-4142, MR 0813071
  12. ^Eric W. Weisstein, Ramanujan Continued Fractions at MathWorld
Sours: https://en.wikipedia.org/wiki/Square_root_of_5

Root 5 square

What is the square root of 5?

All positive numbers normally have two square roots, a positive one and a negative of the same size. We denote the positive (a.k.a. principal) square root of by .

A square root of a number is a number such that . So if then also .

However, popular usage is that "the square root" refers to the positive one.

Suppose we have a positive number which satisfies:

Then multiplying both sides by we get:

Then subtracting from both sides we get:

So we have found:

SInce this continued fraction does not terminate, we can tell that cannot be represented as a terminating fraction - i.e. a rational number. So is an irrational number a little smaller than . For better rational approximations you can terminate the continued fraction after more terms.

For example:

Unpacking these continued fractions can be a little tedious, so I generally prefer to use a different method, namely the limiting ratio of an integer sequence defined recursively.

Define a sequence by:

The first few terms are:

The ratio between terms will tend to .

So we find:

Sours: https://socratic.org/questions/what-is-the-square-root-of-5
Square Root of 5 Simplified

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