p. 1

square root properties square root properties a square root of a number a is a number y such that y2 a or in other words a number y whose square the result of multiplying the number by itself or y × y is a every non-negative real number a has a unique non-negative square root called the principal square root which is denoted by where is called radical sign for example the principal square root of 9 is 3 denoted because 32 3 × 3 9 and 3 is non-negative the term whose root is being considered is known as the radicand the radicand is the number or expression underneath the radical sign in this example 9 every positive number a has two square roots which is positive and which is negative together these two roots are denoted although the principal square root of a positive number is only one of its two square roots the designation the square root is often used to refer to the principal square root know more about transformations on the coordinate plane tutorcircle.com page no 1/4

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p. 2

for positive a the principal square root can also be written in exponent notation as a1/2 square roots of negative numbers can be discussed within the framework of complex numbers more generally square roots can be considered in any context in which a notion of squaring of some mathematical objects is defined including algebras of matrices endomorphism rings etc square roots of positive whole numbers that are not perfect squares are always irrational numbers numbers not expressible as a ratio of two integers that is to say they cannot be written exactly as m/n where m and n are integers this is the theorem euclid x 9 almost certainly due to theaetetus dating back to circa 380 bc the particular case is assumed to date back earlier to the pythagoreans and is traditionally attributed to hippasus it is exactly the length of the diagonal of a square with side length 1 properties the principal square root function usually just referred to as the square root function is a function that maps the set of non-negative real numbers onto itself in geometrical terms the square root function maps the area of a square to its side length the square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares see square root of 2 for proofs that this is an irrational number and quadratic irrational for a proof for all non-square natural numbers the square root function maps rational numbers into algebraic numbers a superset of the rational numbers learn more point of tangency tutorcircle.com page no 2/4

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p. 3

the most common iterative method of square root calculation by hand is known as the babylonian method or heron s method after the first-century greek philosopher heron of alexandria who first described it the method uses the same iterative scheme as the newton-raphson process yields when applied to the function using the fact that its slope at any point is but predates it by many centuries it involves a simple algorithm which results in a number closer to the actual square root each time it is repeated the basic idea is that if x is an overestimate to the square root of a non-negative real number a then will be an underestimate and so the average of these two numbers may reasonably be expected to provide a better approximation though the formal proof of that assertion depends on the inequality of arithmetic and geometric means that shows this average is always an overestimate of the square root as noted below thus assuring convergence to find x start with an arbitrary positive start value x the closer to the square root of a the fewer iterations will be needed to achieve the desired precision replace x by the average between x and a/x that is representing the newton-raphson scheme resulting in tutorcircle.com page no 3/4 page no 2/3

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p. 4

thank you for watching presentation

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