1 Little Notes on Square Root 1
1 Little Notes on Square Root
1.1 Radical
√ √
The symbol is called radical and the expression n x is read as x radical n. The horizontal
√ above x is called vinculum and it indicates that the quantities under it form a
line placed
unit. In n x, n is the index and x is called radicand .
Definition 1.1 (Arithmetic root of degree n of a number a ≥ 0) The n-th root of a non-
negative real number a, where√n is an even number and n ∈ N : n ≥ 2 is a number b such that
bn = a. It is denoted as b = n a.
√ √ p √
Remark 1.1 y 2 = x ⇏ y = x once x = y 2 ⇒ x = y 2 ⇒ x = |y|.
Definition 1.2 (Root of an odd degree n of a number a < 0) An odd degree n is n =
2k + 1, k ∈ N = {1, 2, 3 . . √
.}. Thus, the root an odd√degree n of a negative real number a is a
number b such that b = − n −a and denoted as b = n a.
√
n |a| if n is even
an = (1)
a if n is odd
1.2 Square Root
√ √ √
The special case of n x when n = 2 is written as 2 x = x and read as square√ root of x.
1
Therefore, the square root of a number a is a number b such that b2 = a. Also,
√ a = a 2 for
m n mn b
a ≥ 0. In order to prove it, we can use the fact that (a ) = a . Considering a = a
√ 1
( a)2 = a ⇐⇒ (ab )2 = a1 ⇐⇒ a2b = a1 ⇐⇒ 2b = 1 ⇐⇒ b =
2
√
Let f be the function
√ defined as f : R≥0 → R≥0 , f (x) = x. Note that it does represent
a function because x is defined to be the principal square root and the graph pass the
vertical line test. More precisely, recall that a relation f from X to Y is a subset of X × Y ,
thus f ⊆ X × Y and therefore the elements of f are ordered pairs (x, y). Accordingly, f is a
well defined function if given a set X as domain and Y as the codomain, we can show that:
1.1) f ⊆ X × Y
1.2) The domain of f is X denoted as dom(f ) = X and every element in X is related to some
element in Y , thus ∀x ∈ X, ∃y ∈ Y : (x, y) ∈ f
1.3) Every element of X is related to only one of Y (it does represent the vertical line test).
Thus, ∀x ∈ X, ∀y1 , y2 ∈ Y : (x, y1 ), (x, y2 ) ∈ f ⇒ y1 = y2 . In this case, whenever
(x, y) ∈ f we denote y = f (x).
It is true that domain and codomain are generally defaulted to be subsets of the Real
set. Once f : [0, ∞) → [0, ∞) and it is true that we have a subset f ⊆ [0, ∞) × [0, ∞) such that
1 Little Notes on Square Root
1.1 Radical
√ √
The symbol is called radical and the expression n x is read as x radical n. The horizontal
√ above x is called vinculum and it indicates that the quantities under it form a
line placed
unit. In n x, n is the index and x is called radicand .
Definition 1.1 (Arithmetic root of degree n of a number a ≥ 0) The n-th root of a non-
negative real number a, where√n is an even number and n ∈ N : n ≥ 2 is a number b such that
bn = a. It is denoted as b = n a.
√ √ p √
Remark 1.1 y 2 = x ⇏ y = x once x = y 2 ⇒ x = y 2 ⇒ x = |y|.
Definition 1.2 (Root of an odd degree n of a number a < 0) An odd degree n is n =
2k + 1, k ∈ N = {1, 2, 3 . . √
.}. Thus, the root an odd√degree n of a negative real number a is a
number b such that b = − n −a and denoted as b = n a.
√
n |a| if n is even
an = (1)
a if n is odd
1.2 Square Root
√ √ √
The special case of n x when n = 2 is written as 2 x = x and read as square√ root of x.
1
Therefore, the square root of a number a is a number b such that b2 = a. Also,
√ a = a 2 for
m n mn b
a ≥ 0. In order to prove it, we can use the fact that (a ) = a . Considering a = a
√ 1
( a)2 = a ⇐⇒ (ab )2 = a1 ⇐⇒ a2b = a1 ⇐⇒ 2b = 1 ⇐⇒ b =
2
√
Let f be the function
√ defined as f : R≥0 → R≥0 , f (x) = x. Note that it does represent
a function because x is defined to be the principal square root and the graph pass the
vertical line test. More precisely, recall that a relation f from X to Y is a subset of X × Y ,
thus f ⊆ X × Y and therefore the elements of f are ordered pairs (x, y). Accordingly, f is a
well defined function if given a set X as domain and Y as the codomain, we can show that:
1.1) f ⊆ X × Y
1.2) The domain of f is X denoted as dom(f ) = X and every element in X is related to some
element in Y , thus ∀x ∈ X, ∃y ∈ Y : (x, y) ∈ f
1.3) Every element of X is related to only one of Y (it does represent the vertical line test).
Thus, ∀x ∈ X, ∀y1 , y2 ∈ Y : (x, y1 ), (x, y2 ) ∈ f ⇒ y1 = y2 . In this case, whenever
(x, y) ∈ f we denote y = f (x).
It is true that domain and codomain are generally defaulted to be subsets of the Real
set. Once f : [0, ∞) → [0, ∞) and it is true that we have a subset f ⊆ [0, ∞) × [0, ∞) such that
