If you’ve ever punched “square root of 5” into a calculator and stared at that long decimal, you’re not alone. Most people see 2.2360679… and immediately wonder — what even is this number? Where does it come from? And why doesn’t it stop?
Let’s break it all down. Simple, clear, no unnecessary fluff. You can also use our free square root calculator to instantly find the square root of any number.
What is the Square Root of 5?
The square root of 5 is the number that, when multiplied by itself, gives you exactly 5. In math language, if x² = 5, then x = √5.
The answer is 2.2360679774997896… and it goes on forever. No pattern, no repetition, just endless digits. You can use our square root calculator to get the exact value instantly — along with simplified radical form and step-by-step working.
| Form | Value |
| Decimal | 2.2360679… |
| Simplified Radical | √5 (already simplified) |
| Exponent Form | 5^(1/2) |
| Negative Square Root | -2.2360679… |
Is the Square Root of 5 Rational or Irrational?
This is one of the most searched questions about √5 — and the answer is clear.
The square root of 5 is irrational.
A rational number can be written as a simple fraction — like 1/2 or 7/3. An irrational number cannot. √5 falls in the second category. This is similar to the square root of 2 and the square root of 3, which are also irrational numbers.
Here’s why. If you assume √5 equals some fraction p/q in its simplest form and follow the math through, you hit a contradiction. The proof forces both p and q to be divisible by 5 — but that’s impossible if the fraction is already fully reduced. So the assumption breaks. √5 simply cannot be a fraction.
It never ends, never repeats. That’s the definition of irrational. According to Wikipedia, irrational numbers cannot be expressed as a ratio of two integers — and √5 is a classic example.
Is the Square Root of 5 a Rational Number?
No. As explained above, √5 is not a rational number. It cannot be expressed as p/q where both p and q are integers and q ≠ 0.
5 is not a perfect square. Perfect squares like 4, 9, 16, and 25 have clean whole number square roots. Since 5 sits between the perfect squares 4 and 9, its square root lands somewhere between 2 and 3 — and that “somewhere” is an irrational number.
Square Root of 5 in Decimal Form
In decimal form, the square root of 5 is:
√5 = 2.2360679774997896964091736687747…
For most practical uses, you can round it:
| Rounded To | Value |
| 1 decimal place | 2.2 |
| 2 decimal places | 2.24 |
| 3 decimal places | 2.236 |
| 4 decimal places | 2.2361 |
| 5 decimal places | 2.23607 |
The decimal never terminates and never repeats — which is exactly what makes it irrational. For instant decimal results, try our square root calculator.
Square Root of 5 Simplified
Can √5 be simplified? Short answer — no.
To simplify a square root, you need to find a perfect square factor inside the number. For example, √8 simplifies to 2√2 because 8 = 4 × 2, and √4 = 2.
But 5 is a prime number. It has no perfect square factors other than 1. So √5 is already in its simplest radical form. You can’t break it down further.
√5 = √5 — that’s as simple as it gets.
According to Math is Fun, simplifying square roots requires finding perfect square factors — and since 5 is prime, there are none to find.
How to Calculate Square Root of 5
You don’t need a calculator to get close. There’s an old method — used for thousands of years — called the Babylonian Method or long division method.
Step-by-step:
- Start with a rough guess. Use 2 as your starting point.
- Divide 5 by your guess: 5 ÷ 2 = 2.5
- Average your guess and the result: (2 + 2.5) ÷ 2 = 2.25
- Repeat: 5 ÷ 2.25 = 2.222…
- Average again: (2.25 + 2.222) ÷ 2 = 2.236
After just two rounds, you’re accurate to three decimal places. Each round gets you closer and closer to the real value.
Or just use our free square root calculator and get the answer in one click — with full step-by-step working shown automatically.
Square Root of 5 on the Number Line
√5 sits between √4 = 2 and √9 = 3 on the number line — closer to 2 than to 3.
More precisely, it lands at approximately 2.236.
A simple way to picture it — if you draw a number line and mark 2 and 3, √5 is about 23% of the way from 2 toward 3. Not quite in the middle, but leaning left.
Where Does √5 Appear in Real Life?
More places than you’d expect.
In geometry: Draw a rectangle where one side is 1 unit and the other is 2 units. The diagonal of that rectangle is exactly √5. You can verify it with the Pythagorean theorem: 1² + 2² = 1 + 4 = 5. So the diagonal = √5.
In the golden ratio: The famous golden ratio — roughly 1.618 — is calculated using this formula: (1 + √5) ÷ 2. Every time someone talks about golden ratio proportions in architecture or nature, √5 is quietly powering the math behind it. You can read more about it on Wikipedia’s Golden Ratio page.
In Fibonacci numbers: Binet’s formula, which calculates any Fibonacci number directly, uses √5 right in the middle. Without it, the formula doesn’t work.
Is 5 a Perfect Square?
No, 5 is not a perfect square.
Perfect squares are numbers like 1, 4, 9, 16, 25, 36… — numbers whose square roots are whole integers. Since √5 = 2.236… (not a whole number), 5 is not a perfect square.
It’s actually a prime number, which means it has no factors other than 1 and itself. That’s exactly why its square root can’t be simplified or written as a clean fraction.
Quick Facts About √5
| Property | Value |
| Symbol | √5 |
| Value (decimal) | 2.2360679… |
| Rational or Irrational | Irrational |
| Perfect Square? | No |
| Simplified Form | √5 (cannot simplify) |
| Between | √4 (2) and √9 (3) |
| Used in | Golden ratio, geometry, Fibonacci |
FAQs About Square Root of 5
Q: What is 5 square root of?
The phrase “5 square root of” typically refers to √5, which equals approximately 2.2360679. Use ourcalculator to verify instantly.
Q: What is the square root of 5 squared?
(√5)² = 5. Squaring a square root always gives you back the original number.
Q: Is √5 a real number?
Yes. √5 is a real number. It is irrational, but it is real. Imaginary numbers only come in when you take the square root of a negative number.
Q: What is the square root of -5?
√(-5) = i√5, where i is the imaginary unit. This is a complex number, not a real number.
Q: How do you write square root of 5 in exponent form?
√5 = 5^(1/2) or 5^0.5
Q: How does √5 compare to √2 and √3?
All three are irrational numbers.√2 ≈ 1.414,√3 ≈ 1.732, and √5 ≈ 2.236. Each one is larger than the previous.
Final Thoughts
The square root of 5 is one of those numbers that looks simple on the surface but quietly shows up everywhere — in geometry, in the golden ratio, in Fibonacci sequences. It’s irrational, it can’t be simplified, and its decimal goes on forever.
Want to calculate square roots of other numbers instantly? Try our free Square Root Calculator — it gives you the decimal, simplified radical, and step-by-step working in seconds.
