Answer:
√
−
49
=
7
i
Explanation:
A square root of a number
n
is a number
x
such that
x
2
=
n
Note that if
x
is a Real number then
x
2
≥
0
.
So any square root of
−
49
is not a Real number.
In order to be able to take square roots of negative numbers, we need Complex numbers.
That's where the mysterious number
i
comes into play. This is called the imaginary unit and has the property:
i
2
=
−
1
So
i
is a square root of
−
1
. Note that
−
i
is also a square root of
−
1
, since:
(
−
i
)
2
=
(
−
1
⋅
i
)
2
=
(
−
1
)
2
⋅
i
2
=
1
⋅
(
−
1
)
=
−
1
Then we find:
(
7
i
)
2
=
7
2
⋅
i
2
=
49
⋅
(
−
1
)
=
−
49
So
7
i
is a square root of
−
49
. Note that
−
7
i
is also a square root of
−
49
.
What do we mean by the square root of
−
49
For positive values of
n
, the square root is usually taken to mean the principal square root
√
n
, which is the positive one.
For negative values of
n
, the square roots are both multiples of
i
, so neither positive nor negative, but we can define:
√
n
=
i
√
−
n
With this definition, the principal square root of
−
49
is:
√
−
49
=
i
√
49
=
7
i
Footnote
The question remains: Where does
i
come from?
It is possible to define Complex numbers formally, as pairs of Real numbers with rules for arithmetic like this:
(
a
,
b
)
+
(
c
,
d
)
=
(
a
+
c
,
b
+
d
)
(
a
,
b
)
⋅
(
c
,
d
)
=
(
a
c
−
b
d
,
a
b
+
c
d
)
These rules for addition and multiplication work as expected with commutativity, distributivity, etc.
Then Real numbers are just Complex numbers of the form
(
a
,
0
)
and we find:
(
0
,
1
)
⋅
(
0
,
1
)
=
(
−
1
,
0
)
That is
(
0
,
1
)
is a square root of
(
−
1
,
0
)
Then we can define
i
=
(
0
,
1
)
and:
(
a
,
b
)
=
a
(
1
,
0
)
+
b
(
0
,
1
)
=
a
+
b
i
