How to Calculate and Understand the Square Root of 1
The square root of a number is the value obtained by raising the number to the power ½. The number obtained by multiplying a number by itself is called a square number. Square and square roots are inverse Mathematical operations. Squares and square roots are used generally in solving quadratic equations and many other Mathematical calculations. Square root is denoted by a symbol ‘√’. Square root of a number ‘x’ is written as √x or x½. Square root of any number has two values: one positive and one negative. However, the magnitude of both the values remain the same.
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Value of Root 1 = +1 or -1
Significant Facts About ‘1’
1 is the most important element of Mathematics. One or unity in Mathematics is used to represent a single entity in a number, measurement, or calculation. The number ‘1’ has a few peculiar properties which are very important in Mathematical calculations. They are:
‘1’ is the number used to represent a single identity.
‘1’ is added to any integer to get the immediate successive integer.
When ‘1’ is subtracted from any integer, the immediately preceding integer is obtained.
1 is the multiplicative identity of any number. i.e. When any number is multiplied by itself, the number itself is obtained as the product.
The multiplicative inverse of any number is the value obtained when ‘1’ is divided by the number.
When any number is divided by ‘1’, the answer is the number itself.
When the number is divided by itself, the answer obtained is one.
The value of any number raised to the power zero is equal to unity.
Square Root of +1
It is very important to know how to find the square root of 1 because it gives a clear understanding of finding the square root of other integers. A positive value of one can be written as \[1 \times 1 or 1^{2}\].
So, square root of 1 can be calculated as:
\[\sqrt{1} = \sqrt{1^{2}} = \pm 1\]
The formula for finding the roots of a quadratic equation can also be used to find the square root of 1.
Let the square of the number ‘x’ be equal to ‘1’. This can be written as:
\[x^{2} = 1\]
\[x = \sqrt{1}\] → (1)
The above equation is a quadratic equation which can be represented in standard form as:
\[x^{2} + 0 x - 1 = 0\]
The above equation is of the form ax2 + bx + c = 0. So, a = 1, b = 0 and c = -1.
The value of ‘x’ can be found using the formula:
\[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \]
\[x = \frac{-0 \pm \sqrt{0^{2} - 4x \times 1 \times -1}}{2 \times 1} = \pm \frac{\sqrt{4}}{2} =\pm \frac{2}{2} \rightarrow (2)\]
Comparing equations (1) and (2), we can infer that the value of under root 1 is equal to either positive or negative unity.
Value of root 1 = \[\pm\] 1
Most commonly, the value of under root 1 is taken as positive unity or + 1.
Value of Square Root of -1
Root value of ‘-1’ does not exist in theory. It is an imaginary number represented as ‘i’. Root of -1 is generally used to represent complex numbers which include both the real part and the imaginary part. With the knowledge of the square root of negative unity, the root value of any negative number can be found. Square root of -1 is a positive or negative imaginary unit ‘i’. However, in most cases, the value of the root of -1 is taken as a positive imaginary unit ‘i’.
Square Root of First 30 Integers:
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Square root 1 to 10:
Values of Square Root 1 to 10 is Listed in the Table Below:
These values of square root 1 to 10 are depicted on the number line as a square root spiral.
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Example Problems:
1. Solve for p if \[p^{2} + 8 = 3\]
Solution:
\[p^{2} + 8 = 3\]
\[p^{2} = 3 - 8 \]
\[p^{2} = - 5 \]
\[p = \sqrt{-5} = \sqrt{-1} . \sqrt{5} \]
\[p = \sqrt{5i}\]
2. Find the value of \[7\sqrt{1} - 5\sqrt{1} + 2\sqrt{1}\] using the value of under root 1.
Solution:
Value of \[\sqrt{1} = 1\]
\[7\sqrt{1} - 5\sqrt{1} + 2\sqrt{1}\]
= 7 (1) - 5 (1) + 2 (1)
= 7 - 5 + 2 = 4.
Fun Facts:
‘I’ is the first unit of imaginary numbers. It is equivalent to number ‘1’ in real numbers.
When negative unity is raised to the power of odd numbers the answer is -1 and when negative unity is raised to the power of even numbers, the answer is + 1.
The value of root 1 to any power is equal to 1.
Significance of Square Roots
In the applied area of Mathematics, the concept of square roots is considered to be highly important. The concept lays the basic foundation for algebra. Students who plan to score exceptionally in the subject should study this chapter in detail.
Vedantu tries to explain complex concepts in simple terms. It makes it convenient for the students to dive deeper into the logical reasoning behind the numerical values. There are many benefits for studying square roots-
Square roots from basic to complex hold a significant weightage in board exams.
The tricks related to calculating the square roots help in setting the mind map for mastering Math.
It further helps in taking your mathematical skills to the level of abstraction.
With the help of square roots, students will be able to hone their calculative skills in an intelligent manner.
Besides being important in the concept of algebra, square roots play a significant role in boosting your child's theoretical and statistical methods.
In addition to Math, square roots would help you to get a better understanding of some important laws in Physics.
Learn Square Roots Easily
Square roots might seem to be complicated at times. With Vedantu, Students can clear all their doubts related to it.
In order to make the concept easy, we provide sample problems at the right intervals. You can easily get a firm grip over the topics that are considered to be of main importance in solving algebra.
To start with, students should understand the definition of the concept as defined by the Vedantu experts. The definition is formulated by the experts and will stick with you in the long run.
Before coming to the other numbers, it is important that you take one step at a time. Starting from Number 1, Vedantu has covered all the details related to its value, method and example problems to help you score well on the topic.
Vedantu provides a detailed tabular representation for the square root of the first 30 integers. It also provides a table consisting of values from 1 to 10.
Experts at Vedantu make sure to include all the concepts for the particular topic you are looking for. Along with the square root of +1, it has also covered the square root of -1. Questions related to it are most likely to be asked in the exams. It helps you in scoring well on the 'High-order thinking skills(HOTS).
To make sure that students have fun during their learning process, Vedantu consists of 'fun facts'
related to the topic. Students from all the classes find it intriguing and curious enough to know more about the concept.
To score well in Mathematics, it is very important to keep practicing the example problems. Vedantu experts have formulated some important examples along with the solutions. It will help you in understanding the kind of questions expected out of the topic.
FAQs on What Is the Square Root of 1?
1. What is the square root of 1?
The square root of 1 has two answers: 1 and -1. This is because a square root is a value that, when multiplied by itself, gives the original number. Since (1 × 1) = 1 and (–1 × –1) = 1, both 1 and -1 are the square roots of 1. However, when asked for the principal square root, the positive value, 1, is typically used.
2. Why does the number 1 have both a positive and a negative square root?
This is due to the rules of multiplication with signs. Multiplying two positive numbers results in a positive product, and multiplying two negative numbers also results in a positive product. Therefore:
- (+1) × (+1) = 1
- (–1) × (–1) = 1
Because both operations yield 1, both +1 and -1 are considered valid square roots. This principle applies to all positive numbers, not just 1.
3. What is the main difference between a 'square' and a 'square root'?
The main difference lies in the operation they represent. A square is the result of multiplying a number by itself (e.g., the square of 3 is 3 × 3 = 9). A square root is the inverse operation; it is the number that you multiply by itself to get the original number (e.g., the square root of 9 is 3). Essentially, squaring a number makes it larger (if >1), while finding the square root makes it smaller.
4. How is the number 1 an example of a perfect square?
A perfect square is a number that is the product of an integer multiplied by itself. The number 1 is a perfect square because it is the product of the integer 1 multiplied by itself (1 × 1 = 1). Other examples of perfect squares include 4 (from 2 × 2), 9 (from 3 × 3), and 16 (from 4 × 4).
5. How can we find the square root of 1 using the long division method?
The long division method for finding the square root of 1 is very straightforward:
- Step 1: Draw the long division symbol and place '1' inside as the dividend. Since 1 is a single digit, we don't need to pair digits.
- Step 2: Find a number that, when multiplied by itself, is less than or equal to 1. That number is 1.
- Step 3: Write '1' as both the divisor and the quotient.
- Step 4: Multiply the divisor (1) by the quotient (1) to get 1. Write this below the dividend and subtract.
- Step 5: The remainder is 1 - 1 = 0. Since the remainder is zero, the process is complete. The quotient, 1, is the square root of 1.
6. Is it possible to find the square root of -1?
Within the set of real numbers (which are used in most school-level mathematics), you cannot find the square root of -1. This is because multiplying any real number (whether positive or negative) by itself always results in a positive number. However, in advanced mathematics, the square root of -1 is defined and is called an imaginary number, represented by the symbol 'i'. This concept is fundamental to complex numbers, typically introduced in Class 11.
7. How does the square root of 0 compare to the square root of 1?
The primary difference is in the number of solutions.
- The square root of 1 has two distinct solutions: +1 and -1.
- The square root of 0 has only one solution: 0 (since 0 × 0 = 0).
Zero is unique because it is neither positive nor negative, so there is no separate '+0' and '-0'. Therefore, while every positive number has two real square roots, zero has only one.
