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The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root § Principal square root of a complex number.
A square root of a number x is a number r which, when squared, becomes x: =. Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
Radical expression involving roots, also known as an nth root; Radical symbol (√), used to indicate the square root and other roots; Radical of an algebraic group, a concept in algebraic group theory; Radical of an ideal, an important concept in abstract algebra; Radical of a ring, an ideal of "bad" elements of a ring
In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one.
There is no known polynomial-time algorithm for computing the square-free part of an integer. [ 3 ] The definition is generalized to the largest t {\displaystyle t} -free divisor of n {\displaystyle n} , r a d t {\displaystyle \mathrm {rad} _{t}} , which are multiplicative functions which act on prime powers as
The dictionary was edited by the honorary director general of the board Maulvi Abdul Haq who had already been working on an Urdu dictionary since the establishment of the Urdu Dictionary Board, Karachi, in 1958. [1] [2] [3] Urdu Lughat consists of 22 volumes. In 2019, the board prepared a short concise version of the dictionary in 2 volumes.
Feroz-ul-Lughat Urdu Jamia (Urdu: فیروز الغات اردو جامع) is an Urdu-to-Urdu dictionary published by Ferozsons (Private) Limited. It was originally compiled by Maulvi Ferozeuddin in 1897. The dictionary contains about 100,000 ancient and popular words, compounds, derivatives, idioms, proverbs, and modern scientific, literary ...