Square Root Calculator .Co

Unmatched math delimiters. Adding final one for you.

The simple answer is: yes you can get negative numbers out of square roots. In fact, should you wish to find the square root of any positive real numbers, you will get two results: the positive and negative versions of the same number.

Writing a Square Root Equation for Positive and Negative Results

16 = 4 * 4 = (-4) * (-4)

In the equation above, you can either multiply 4 by itself or multiply (-4) by itself to get the result of 16. Thus, the square root of 16 would be:

The √ symbol is called the radical symbol, while the number or expression inside the symbol—in this case, 16—is called a radicand.

Why would we need a ± symbol in front of 4? Well, as we have discussed before, the square roots of 16 can be either 4 or (-4)​.

Most people would simply write this equation simply as . While it is technically true and there’s nothing wrong with it, it doesn’t tell the whole story.

Instead, you can also write the equation in such a way that it explicitly indicates that you want both the positive and negative square root’s results:

This way, other people can easily tell that the one who writes the equation wishes to have positive and negative numbers as the result.

Perfect and Imperfect squares

As you may know, 16 is a perfect square. Perfect squares are radicands in the form of an integer, or a whole number, that has a square root of another integer. In the example above, 16 is a perfect square because it has the number 4 as its square root.

Positive real numbers are not always perfect squares. There are also other numbers such as 3, 5, or 13 that are referred to as imperfect squares. If a radicand is not a perfect square, then the square root of the radicand won’t result is an integer. Take a look at the equation below.

5 is not a perfect square. Therefore, its square root won’t be an integer. Additionally, the square root is not even a rational number. Rational numbers are numbers that can be expressed as fractions composed of two integers, e.g. 7/2​, 50/4​, and 100/3​.

Square Roots of Zero and Negative Numbers

We mentioned earlier that any positive real numbers have two square roots, the positive one and the negative one. What about negative numbers and zero?

For zero, it only has one square root, which is itself, 0.

On the other hand, negative numbers don’t have any real square roots. Any real number—whether it’s positive or negative—that is multiplied by itself is always equal to a positive number, except for 0. Instead, the square root of all negative numbers is an imaginary number.

By definition, the square root of (-1)​ is i​, which is an imaginary unit. As a side note, imaginary numbers do not have a tangible value. They are not part of real numbers in the sense that they can’t be quantified on the number line. However, they are still used in math and the study of sciences including quantum mechanics, electricity, and more.

To get a better understanding, let’s take a look at an example. For instance, let’s say we want to identify the square root of (-9)​, what would it be?

Therefore, ​​ is equal to 3i​. This result can also be written as ​

How about the square root of ​(-3)?

Thus, ​​ is equal to ​​. Or, it can also be written as ​-3 = 3i2.

Conclusion

If you want to get a negative out of a square root, just remember to write your square root equation using the ± symbol. This will let you and others know that you are looking for it — and it is in there!

Square roots of negative numbers were
introduced in a prior lesson:


Arithmetic with Complex Numbers
in the Algebra II materials.


For your convenience, that material is repeated
(and extended) here.

Firstly, recall some information
from beginning algebra:

The Square Root of a Nonnegative Real Number

For a nonnegative real number $\,k\,,$

Thus, the number that gets to be
called the square root of $\,k\,$
satisfies two properties:

• it is nonnegative (not negative; i.e., greater than or equal to zero)
• when squared, it gives $\,k\,$

• $\,3\,$ is nonnegative
• 
  $\,3\,,$ when squared,

   
  $\,3^2 = 9$

The Square Root of a Negative Number

THE SQUARE ROOT OF A NEGATIVE NUMBER

Let $\,p\,$ be a positive real number,
so that $\,-p\,$ is a negative real number.

Some of my students like
to think of it this way:


You can slide a minus sign out of a square root,
and in the process, it turns into the
imaginary

Examples

Here are some examples:

• 
  It is conventional to write $\,2i\,,$
  not $\,i2\,.$
  
  
  When there’s no possible misinterpretation
  (see below), write a real number multiplier
  before the $\,i\,.$
  

Two Different Questions;
Two Different Answers

Recall these two different questions,
with two different answers:

• 
  What are the solutions to the
  equation
  $\,x^2 = 4\,$?
  

  $\,x= \pm\ 2\,.$
  
  There are two different
  real numbers which, when squared,

Similarly, there are two different
questions involving complex numbers,
with two different answers:

• 
  What are the (complex) solutions
  to the equation
  $\,x^2 = -4\,$?
  

  Verifying the second solution:

  $$
  \cssId{s74}{(-2i)(-2i)}
  \cssId{s75}{= 4i^2}
  \cssId{s76}{= 4(-1)}
  \cssId{s77}{= -4}
  $$

Square Roots of Products

Does this property work for negative numbers, too?
The answer is no, as shown next.

Here is the correct approach:

Be careful about this!

---

√ = ?

If you enter SQRT(-2) into a spreadsheet on your computer, you will get an error message saying something like «argument must be greater
or equal to 0» because the square root of negative 2 is not possible. There is no real number multiplied by itself that will equal -2.

However, it is possible to calculate the square root of negative 2 with a complex or imaginary square root number. We start by making the imaginary
square root of negative 1. With this, we can combine real and imaginary square roots to make this true:

√ = √ × √

The square root of 2 (rounded to the nearest thousandth) is 1.414. Furthermore, the square root of negative 1 is an imaginary insignificant number (iota) which can be transliterated as i.
That’s it. Now, we have our answer to the square root of negative 2:

√ = 1.414 i

Note: since negative times negative equals positive, one could therefore conclude that -1.414 i is also a correct answer to the square root of negative 2.

Square Root of a Negative Number
Please enter another negative number in the box below to get the square root of that negative number.

Square Root of Negative 3
Here is the next square root problem on our list that we have calculated for you.

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(fill in either box above to explore more about square roots)
Answers are rounded to seven decimal places.

• positive numbers: 2
• negative numbers: -3
• imaginary numbers: -7i
• complex numbers: 8+9i

Notes:

• The square root calculator provides the principal square root (the positive square root which is most commonly used).  
  Multiplying the principal square root by -1 will provide the negative square root if needed.
• The square root of a negative number results in an imaginary number noted by the letter «i».

Square Root:

The square root of a number is a value that when multiplied by itself equals the original number.  
Example:   The square root of 9 = 3.   Thus: 3 x 3 = 9.

Where A and B are valid numbers:
A is the square root of B if:   A x A = B

• Square Root Symbol / Square Root Sign:

  The square root of a number N is commonly represented by:  

  It can also be represented by:  

  In exponential notation, it can be represented by:   N^1/2
  
  «Square root» can be abbreviated as «sqrt»   Example:   sqrt(2) = 1.4142136

  See: square root symbol if you need to add the symbol to a text document.

• How are square roots used:

  Square roots can be used:

  • to solve for the distance between two points (Pythagorean Theorem)
  • to solve for the length of a side of a right triangle (Pythagorean Theorem)
  • to find the solutions to quadratic equations
  • to find normal distribution
  • to find standard deviation
  • basically to solve for a squared variable in an equation

  If you see an equation like N² = 27 , you can solve for N by taking the square root of 27 which
  gives N = 5.1961524

How to Calculate the Square Root of:

1. A Positive Number:
   
       The square root of a positive number can be calculated by:
   <!—
   1. SquareRootCalculator.Co website

   —>

   2. square root calculator above
   3. handheld calculator
   4. spreadsheet
   5. square roots without a calculator

      
      

2. A Negative Number:
   
   To calculate the square root of a negative number, find the square root of the same positive number and multiply by «i». ( where i represents an imaginary number and i = square root of -1)

   Example: square root of -5
   
   = (square root of 5) x (square root of -1)
   
   = (square root of 5) x (i)
   
   = 2.236068 x i
   
   = 2.236068i
   
   

3. An Imaginary Number:
   
   To calculate the square root of an imaginary number, find the square root of the number as if it were a real number (without the i) and then multiply by the square root of i (where the square root of i = 0.7071068 + 0.7071068i)

   Example: square root of 5i
   
   = (square root of 5) x (square root of i)
   
   = (2.236068) x (0.7071068 + 0.7071068i)
   
   = 1.5811388 + 1.5811388i
   
   
   

4. A Complex Number:
   
   To find the square root of a complex number, use the following equation:

   Square root (a + bi) = c + di
   
   Where:
   c = (1/square root of 2) x square root of [ (square root of(a²+b²)) + a ]
   
   d = (sign of b/square root of 2) x square root of [ (square root of(a²+b²)) — a ]

   Example: the square root of 3-5i = c + di
   
   c = (1/square root of 2) x square root of [ (square root of(3²+(-5)²)) + 3 ]
   
   c = 2.1013034
   d = (-1/square root of 2) x square root of [ (square root of(3²+(-5)²)) — 3 ]
   d = -1.1897378
   Thus: square root of 3-5i = 2.1013034 — 1.1897378i

Need to calculate the square root of a number with a handheld calculator or looking to buy a new calculator?

Most handheld calculators have a square root function.
Below is a review of some that are recommended and how to use them.

Graphing Calculator:

• Texas Instruments TI-84 Plus Graphing Calculator

  

  <!—
  
  —>

  To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.

  (Note: This same method will also work with TI-83 and TI-81 calculators)

  (To see what the graph looks like on this calculator,
  click the show graph button below the calculator image on this page.)

  Pros:
  It can be used on many college entrance exams (check your exam criteria).

  It is a popular calculator. (If you need help, the likelihood of finding someone who knows how to use it is higher).

  The display shows seven lines of input/output. Long equations can be viewed and checked. (This is a nice benefit that graphing calculators have compared to scientific calculators that may only have a single line display.) Another benefit of having a large display is you can compare your current answer to past answers that are still on the screen. This can often help you discover an input error that may have otherwise gone unnoticed.

  Cons:
  It is bulkier than a scientific calculator.
  It costs around $85 (USD) more than a scientific calculator.

  Price:
  Best price for this calculator as of 9-2-2014 is around $94.00 (USD).

• Casio Graphing Calculator (fx-9750GII)

  

  <!—

  Click to show y =
  √x
  graph

  —>

  <!— —>

  To take the square root of a number, press [SHIFT] and then [] (the radical symbol is above the x² key) and then the number that you want to find the square root of and then the [EXE] key.

  (To see this graph, click the show graph button below the calculator image on this page.)

  Pros:
  The cost is half the price of a TI-84 calculator.

  It is a little smaller than the TI-84 calculator.

  It can be used on many college entrance exams (check your exam criteria).

  The display shows seven lines of input/output.

  Cons:

  It is not as popular as the TI-84 calculator. (It may be harder to find someone to help if you have a question about how to use the calculator.)

  Price:
  Best price as of 9-2-2014 is around $42.74 (USD).

Scientific Calculator:

• Texas Instruments TI-36X Pro Scientific Calculator

  

  To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.
  Note:  If your calculator is in «Math Print» mode, the display will show as the answer. To convert this answer to a decimal number, press the [<>~] key (the «Answer Toggle» key) which is the key above the [ENTER] key.

  Pros:
  It is good if a graphing calculator is not permitted.
  It is smaller and lighter than a graphing calculator.
  The display shows four lines of input/output. Some scientific calculators only show one line of input/output.

  Cons:
  The symbols shown on the function keys ( + , — , etc.) may be hard to see because they are the same silver color as the keys.

  Price:
  Best price for this calculator on 9-2-2014 is around $19.99 (USD).

Math Help / Math Tutor

If you need help with math:

• There are a lot of online resources that are free to use and are available when you need them. Some sites have online courses, instructional videos, and even math games to help you learn more about math.
• Ask someone who is good at math to help. If you can’t find anyone who can help, a math teacher at a local school may be able to help or recommend a math tutor in your area.
• A local library may be able to help not only with informational books but they may be able to recommend a math tutor also.
• There are several websites that can help you find a local tutor if you would like in-person tutoring. Also, some tutors are able to help from a remote location if needed. Prices are typically between $10 to $50 (USD) per hour. Always research and try to find reviews, ratings, or feedback from others who have used a particular tutor before making a selection.