Square Root of -113

The square root is the inverse of the square of the number. Since -113 is negative, its square root is not a real number. The square root of -113 is expressed in terms of imaginary numbers. In the radical form, it is expressed as √-113, whereas in exponential form, it is (-113)^(1/2). In terms of imaginary numbers, it is expressed as √113 * i, which is an imaginary number because it involves the imaginary unit i, where i = √-1.

For negative numbers, the square root involves imaginary numbers. The concept of imaginary numbers is used to express square roots of negative numbers. Let us now learn how to find the square root of -113 using this concept:

• Imaginary unit method
• Approximation method for √113

The imaginary unit is used to express the square roots of negative numbers. The imaginary unit is denoted by i, where i = √-1.

Step 1: Express -113 as the product of 113 and -1: -113 = 113 × (-1)

Step 2: Use the property of square roots to separate: √-113 = √(113 × -1) = √113 × √-1

Step 3: Replace √-1 with i: √-113 = √113 × i Thus, the square root of -113 is √113 * i.

Approximation method can be used to estimate the square root of the positive part, 113.

Step 1: Find the closest perfect squares around 113. The closest perfect squares are 100 (10^2) and 121 (11^2). So, √113 falls between 10 and 11.

Step 2: Linear approximation formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square) For 113: (113 - 100) ÷ (121 - 100) = 13 ÷ 21 ≈ 0.619

Step 3: Add this to the smaller root: 10 + 0.619 ≈ 10.619 Therefore, √113 ≈ 10.619. Thus, the square root of -113 is approximately 10.619i.

• Square root: The inverse of squaring a number. For negative numbers, involves imaginary numbers.
   
• Imaginary unit: Denoted by i, it is the square root of -1, used to express square roots of negative numbers.
   
• Complex number: A number of the form a + bi, where a and b are real numbers, and i is the imaginary unit.
   
• Magnitude: The absolute value of a complex number, calculated as √(a^2 + b^2).
   
• Complex conjugate: For a complex number a + bi, its conjugate is a - bi, reflecting it across the real axis.