没有计算器如何求平方根？

数字系统是为不同数字及其排列方式定义的系统。有许多类型的数字系统，但大多数是众所周知的 4 种类型。它们是二进制数字系统、十进制数字系统、八进制数字系统和十六进制数字系统。十进制数系统主要用于数学，它涉及从 0 到 9 的数字。对数字进行了多种操作，例如求数字的平方和平方根，让我们详细了解数字的平方根，

平方根

一个数字的平方根是一个值，当它与自身相乘时会得到原始数字。例如，9的平方根是3，3乘以它自己，得到的原始数是9。数学上表示平方根的符号是√。

这个符号（√）称为部首，部首符号内的数字称为radicand。根符号内的数字或值可能是完美的正方形或不完美的正方形。例如 - 4 是一个完美的正方形，而 3 是一个不完美的正方形。因此，根据根内值的性质，最终答案或平方根可能是十进制数的自然数。
现在让我们找出如何计算不同数字的平方根。

没有计算器的平方根

如上所定义，数字的平方根是与自身相乘时仅提供原始数字的值。不用计算器求平方根的三种方法

质因数分解

这是一个很长但很简单的求任何数平方根的方法。素数分解涉及找到该数字的因数，然后将公共数字配对为一对两个。最后，取主要因素的平方根。让我们看一个这样的例子，

问题：求484的平方根

解决方案：

484=2 × 2 × 11 × 11

So, √484= √(2 × 2 × 11 × 11) = 2 × 11 =22

猜测和检查方法

此方法用于给出任何数字的近似值。猜测方法可以节省时间，因为它给出了根存在的近似值范围。当根内的数字是不完美的数字时，效率更高。让我们看一个这样的例子，

问题：求20的平方根。

解决方案：

Start guess and check method by noting that since √16 = 4 and √25 = 5, then √20 must be between 4 and 5. As second step, in order to reach nearer to the actual answer, lets take a number between 4 and 5. lets assume it to be 4.5. Lets do square of 4.5 which comes out to be 20.25, which is greater than 20, therefore the root must be smaller than 4.5, lets choose 4.4, square of 4.4 is 19.36. thus, the most approx and accurate root of 20 is 4.4

长除法

这是获得不完美平方的平方根的一种非常简单的方法。长除法比其他方法更受欢迎，因为它提供了准确的答案。让我们用一个例子来理解这个算法，

问题：求627的平方根

Solution:

Step 1 Group the numbers in pairs from right to left ,leaving one or two digit in left (here its 6).

Step 2 Think of a number whose square is less than the first number (6), its 2, So,write it like this –
                 

Step 3 Is to square the number 2 and write the result beneath 6 and then subtract as shown below,
           

Step 4  Multiply the quotient by 3 and and write it down in parenthesis with an empty line next to it as shown below,
          

Step 5 Now find out the number which when multiplied by forty something would be lesser than 225. Lets guess 5. then            45×5=225, which is less than 227, So write it as shown below-
          

Step 6 Then repeating step 4, multiply the quotient with 2 write it down in parenthesis with an empty line next to it as shown below,
         

Step 7 Repeating step 5, find out the number which when multiplied by five hundred something would be lesser than 2000. Lets guess 5, then 505×5=2525, which is bigger than 2000, lets guess 4, then 504×3=1512. So write it as shown below,
   

The square root of 627 with two decimal place is 25.03, which is accurate.

示例问题

问题 1：求 144 的平方根

解决方案：

144=2 × 2 × 2 × 2 × 3 × 3

So √144= √(2 × 2 × 2 × 2 × 3 × 3) = 2 × 2 × 3 =12

问题 2：求 169 的平方根

解决方案：

169=13 × 13

So √144= √(13 × 13) = 13

问题 3：通过猜测和检查方法找到 6 的平方根。

解决方案：

Start guess and check the method by noting that since √9 =3 and √4 = 2, then √6 must be between 2 and 3. As the second step, in order to reach nearer to the actual answer, let’s take a number between 2 and 3. Let’s assume it to be 2.5. Let’s do a square of 2.5 which comes out to be 6.25, which is greater than 6. therefore the root must be smaller than 2.5. Let’s choose 2.4, square of 2.4 is 15.76. Thus, the most approx and accurate root of 6 is 2.4