What is the irrational number of 58?

Wyatt,

Your question is a bit ambiguous....  In the answers below, George has used calculus to iteratively estimate and then drive closer to the answer (this is the fundamental procedure of Newton's method which Robert also demonstrated in the comment-answer).  Roman has provided a somewhat complicated way to calculate square roots by hand.  Both are valid and correct methods, and both provided approximations to the calculator decimal answer of  7.615773105863909...  (The true answer written in decimal form never ends and never repeats, by the way, which means it is an irrational number.)

However, there is another answer that you may be expecting, or if not, you are likely to use in the future.  This answer preserves the exact true value of the answer, but does not usually eliminate the radical (it only does this if the number is a perfect square).  This is called "simplifying the radical" and is performed like this:

To simplify the square root of a number N, start from i=2, and decide whether i2 is evenly dividable into N.  Each time you find that it is, you re-assign N/i2 to N, and continue the process, starting from the same i you just found.  If i2 is not evenly divisible into N, then you move on to the next prime number.  Once you get to the point where i2 is greater than N, or if N=1, or if N is prime, then you stop.  Then, all of the i's you took out get multiplied together, and that gets multiplied by the remaining square root of N.

Here's how to do this for N = 96 (not the number you are asking about):

N=96 i=2  96/4 = 24 -> Take out a 2N=24 i=2  24/4 = 6  -> Take out a 2 (again)N=6  i=2  6/4       -> Doesn't divide evenly

N=6  i=3  9 > 6     -> STOP

Now you multiply all the i's you took out, and multiply that by the square root of what was left.  So the sqrt(96) = 2 * 2 * sqrt(6) = 4 * sqrt(6)

Here's a more complicated example - sqrt(19800):

N=19800 i=2 19800/4  = 4950 -> Take out 2N= 4950 i=2  4950/4         -> Doesn't divide evenlyN= 4950 i=3  4950/9  =  550 -> Take out 3N=  550 i=3   550/9         -> Doesn't divide evenlyN=  550 i=5   550/25 =   22 -> Take out 5

N=   22 i=5    22/25        -> i2 > N - STOP

Therefore, sqrt(19800) = 2 · 3 · 5 · sqrt(22) = 30·sqrt(22)

Your question, sqrt(58) is much simpler, because it is the product of two primes, 2 and 29, and so nothing can be "taken out", and thus cannot be reduced.

Is 58 an Irrational Number?

Is 59 an Irrational Number?

In this article we're going to calculate the square root of 58 and explore what the square root is and answer some of the common questions you might. We'll also look at the different methods for calculating the square root of 58 (both with and without a computer/calculator).

Square Root of 58 Definition

In mathematical form we can show the square root of 58 using the radical sign, like this: √58. This is usually referred to as the square root of 58 in radical form.

So what is the square root? In this case, the square root of 58 is the quantity (which we will call q) that when multiplied by itself, will equal 58.

√58 = q × q = q2

Is 58 a Perfect Square?

In math, we refer to 58 being a perfect square if the square root of 58 is a whole number.

In this case, as we will see in the calculations below, we can see that 58 is not a perfect square.

To find out more about perfect squares, you can read about them and look at a list of 1000 of them in our What is a Perfect Square? article.

Is The Square Root of 58 Rational or Irrational?

A common question is to ask whether the square root of 58 is rational or irrational. Rational numbers can be written as a fraction and irrational numbers cannot.

A quick way to check this is to see if 58 is a perfect square. If it is, then it is a rational number. If it's not a perfect square then it's an irrational number.

We already know if 58 is a perfect square so we also can see that √58 is an irrational number.

Can the Square Root of 58 Be Simplified?

58 can be simplified only if you can make 58 inside the radical symbol smaller. This is a process that is called simplifying the surd. In this example square root of 58 cannot be simplified.

√58 is already in its simplest radical form.

How to Calculate The Square Root of 58 with a Calculator

If you have a calculator then the simplest way to calculate the square root of 58 is to use that calculator. On most calculators you can do this by typing in 58 and then pressing the √x key. You should get the following result:

√58 ≈ 7.6158

How to Calculate the Square Root of 58 with a Computer

On a computer you can also calculate the square root of 58 using Excel, Numbers, or Google Sheets and the SQRT function, like so:

SQRT(58) ≈ 7.615773105864

What is the Square Root of 58 Rounded?

Sometimes you might need to round the square root of 58 down to a certain number of decimal places. Here are the solutions to that, if needed.

10th: √58 ≈ 7.6

100th: √58 ≈ 7.62

1000th: √58 ≈ 7.616

What is the Square Root of 58 as a Fraction?

We covered earlier in this article that only a rational number can be written as a fraction, and irrational numbers cannot.

Like we said above, since the square root of 58 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 58 rounded to the nearest hundredth.

√58

≈ 7.6/1

≈ 762/100

≈ 7 31/50

What is the Square Root of 58 Written with an Exponent?

All square root calculations can be converted to a number (called the base) with a fractional exponent. Let's see how to do that with the square root of 58:

√b = b½

√58 = 58½

How to Find the Square Root of 58 Using Long Division

Finally, we can use the long division method to calculate the square root of 58. This is very useful for long division test problems and was how mathematicians would calculate the square root of a number before calculators and computers were invented.

Step 1

Set up 58 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

Step 2

Starting with the first set: the largest perfect square less than or equal to 58 is 49, and the square root of 49 is 7 . Therefore, put 7 on top and 49 at the bottom like this:

Step 3

Calculate 58 minus 49 and put the difference below. Then move down the next set of numbers.

Step 4

Double the number in green on top: 7 × 2 = 14. Then, use 14 and the bottom number to make this problem:

14? × ? ≤ 900

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 6.

Now, enter 6 on top:

That's it! The answer shown at the top in green. The square root of 58 with one digit decimal accuracy is 7.6. Notice that the last two steps actually repeat the previous two. To add decimal places to your answe you can simply add more sets of 00 and repeat the last two steps.

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√58 = q × q = q2

√58 is an irrational number

√58 is already in its simplest radical form.

√58 ≈ 7.615773106

SQRT(58) ≈ 7.6157731058639

10th: √58 ≈ 7.6

100th: √58 ≈ 7.62

1000th: √58 ≈ 7.616

√58

≈ 7 31/50

√b = b½

√58 = 58½

Step 1)

14? × ? ≤ 900

We often refer to perfect square roots on this page. You may want to use the list of perfect squares for reference.