What Is a Square Root?
A square root is a number that, when multiplied by itself, gives the original value. For example, 7 × 7 = 49, so 49 = 7.
Every positive number has two roots (±), but calculators usually display the principal square root, the non-negative one. This Racine Carree Calculator shows that principal value clearly.
How to Calculate Square Roots with the Racine Carree Calculator
Use this online square root calculator to solve real and complex roots in seconds:
Step 1: Enter any number
Type a positive number like 64, a decimal like 2.25, or a negative number like -16.
Step 2: Read the principal result
For non-negative inputs, the calculator returns the real principal root, such as 64 = 8.
Step 3: Check simplified and polar forms
You also get simplified radical notation and polar form, which is useful for math classes and engineering contexts.
Step 4: Understand negative roots
When input is negative, the result includes i (the imaginary unit), because -1 = i.
Example: Square Root of a Negative Number
To evaluate -81, separate the negative sign and use the imaginary unit:
- Rewrite: -81 = 81·-1.
- Compute real part: 81 = 9.
- Use definition: -1 = i.
- Final answer: -81 = 9i.
Popular Square Root Queries
These long-tail topics are commonly searched by learners using a Racine Carree Calculator:
How to find the square root of a negative number?
Rewrite -a as a·-1, then replace -1 with i. For example, -81 = 9i.
How to simplify square roots step by step?
Factor out perfect squares first. Example: 72 = 36×2 = 62.
Is there a square root calculator with complex number explanation?
Yes. This calculator shows the principal result, simplified form, polar form, and a short explanation for complex roots.
What is the difference between principal square root and ± root?
The ± symbol refers to two solutions in equations, while the principal square root is the single default non-negative value.
What is the imaginary unit i in square roots?
i is defined by i² = -1, allowing square roots of negative numbers to be expressed in the complex number system.
FAQ: Square Roots, Complex Numbers, and i
Why does a negative number produce i?
Because no real number squares to a negative value. In complex numbers, we define i by i² = -1, so -a = ai for a>0.
What is the principal square root?
It is the non-negative root returned by default in calculators. For example, the principal root of 25 is 5, even though both 5 and -5 satisfy x² = 25.
Can I use decimals in this calculator?
Yes. You can enter integers or decimals, and the calculator will show a formatted result instantly.
What is the difference between simplified form and polar form?
Simplified form keeps radicals compact (like 32), while polar form expresses complex numbers with cosine and sine terms.