1) Complex numbers are written in the form a + bi, where a is the real part and bi is the imaginary part.
2) Complex numbers can be added, subtracted, multiplied, and divided using standard algebraic rules, resulting in another complex number.
3) When taking the square root of a negative number or solving quadratic equations, complex number solutions may arise with real and imaginary parts.
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Introduction
Johnny Depp is an actor known for his chameleon-like ability to transform into a wide range of characters. from the eccentric Captain Jack Sparrow in "Pirates of the Caribbean" to the introspective Edward Scissorhands. His long hair is one constant throughout his evolving roles and public appearances. Johnny Depp long hair is not a style choice but a significant aspect of his identity. contributing to his allure and mystique. This article explores the journey and significance of Johnny Depp long hair. highlighting how it has become integral to his brand.
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1980s: The Rise of a Young Heartthrob
Johnny Depp's journey in Hollywood began in the 1980s. with his breakout role in the television series "21 Jump Street." During this time, his hair was short, but it was already clear that Depp had a penchant for unique and edgy styles. By the decade's end, Depp started experimenting with longer hair. setting the stage for a lifelong signature.
1990s: From Heartthrob to Icon
The 1990s were transformative for Johnny Depp his career and personal style. Films like "Edward Scissorhands" (1990) and "Benny & Joon" (1993) saw Depp sporting various hair lengths and styles. But, his long, unkempt hair in "What's Eating Gilbert Grape" (1993) began to draw significant attention. This period marked the beginning of Johnny Depp long hair. which became a defining feature of his image.
The Iconic Roles: Hair as a Character Element
Edward Scissorhands (1990)
In "Edward Scissorhands," Johnny Depp's character had a wild and mane that complemented his ethereal and misunderstood persona. This role showcased how long hair Johnny Depp could enhance a character's depth and mystery.
Captain Jack Sparrow: The Pirate with Flowing Locks
One of Johnny Depp's iconic roles is Captain Jack Sparrow from the "Pirates of the Caribbean" series. Sparrow's long, dreadlocked hair symbolised his rebellious and unpredictable nature. The character's look, complete with beads and trinkets woven into his hair. was a collaboration between Depp and the film's costume designers. This style became iconic and influenced fashion trends and Halloween costumes worldwide.
Other Memorable Characters
Depp's long hair has also been featured in other roles, such as Ichabod Crane in "Sleepy Hollow" (1999). and Roux in "Chocolat" (2000). In these films, his hair added a layer of authenticity and depth to his characters. proving that Johnny Depp with long hair is more than a style—it's a storytelling tool.
Off-Screen Influenc
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2. When we take the square root of both sides of an
equation or use the quadratic formula, sometimes we
get a negative under the square root. Because of this,
we'll introduce the set of complex numbers.
i = −1
2
This is called the imaginary unit and its square is -1.
We write complex numbers in standard form and they look like:
a + bi
This is called the real part This is called the imaginary part
3. We can add, subtract, multiply or divide complex numbers. After
performing these operations if we’ve simplified everything correctly we
should always again get a complex number (although the real or
imaginary parts may be zero). Below is an example of each.
Combine real parts and
ADDING (3 – 2i) + (5 – 4i) = 8 – 6i combine imaginary parts
Be sure to distribute the
SUBTRACTING (3 – 2i) - (5 – 4i) negative through before
combining real parts and
3 – 2i - 5 + 4i = -2 +2i imaginary parts
FOIL and then combine like
MULTIPLYING (3 – 2i) (5 – 4i) terms. Remember i 2 = -1
= 15 – 12i – 10i+8i2 Notice when I’m done simplifying
that I only have two terms, a real
=15 – 22i +8(-1) = 7 – 22i term and an imaginary one. If I
have more than that, I need to
simplify more.
4. 3 − 2i 5 + 4i = + 12i − − i − ( 8i )
1515 + 12i1010i8−− 12
DIVIDING
⋅ =
5 − 4i 5 + 4i + + i − − i − −( −i )
25252020i2020i161612
FOIL Combine like terms
i 2 = −1
To divide complex numbers, you multiply the top and
bottom of the fraction by the conjugate of the bottom.
23 + 2i 23 2 This means the same
= = + i complex number, but
41 41 41 with opposite sign on
the imaginary term
We’ll put the 41 under each term so we can
see the real part and the imaginary part
5. Let’s solve a couple of equations that have complex
solutions.
Square root and
x + 25 = 0
2
x = ± − 25
2
don’t forget the ±
-25 -25
x = ± 25( − 1) = ± 25 − 1 = ±5 i
The negative 1
under the square
root becomes i
Use the
x − 6 x + 13 = 0
2
− b ± b 2 − 4ac
quadratic
formula
x=
2a
− ( − 6) ± ( − 6) − 4(1)(13)
2
6 ± 36 − 52
x= =
2(1) 2
6 ± − 16 6 ± 16 i 6±4i
= = = = 3 ± 2i
2 2 2
6. Powers of i
We could continue but notice
i=i that they repeat every group
i = −1
2 of 4.
For every i 4 it will = 1
i = i i = −1(i ) = −i
3 2
To simplify higher powers
i = i i = ( − 1)( − 1) = 1 i and see what is left.
4 2 2 of i then, we'll group all the
4ths
i = i i = 1( i ) = i
5 4
i = ( i ) i = (1) i = i
33 4 8 8
i = i i = 1( − 1) = −1
6 4 2
4 will go into 33 8 times with 1 left.
i = i i = 1( − i ) = −i
7 4 3
i = ( i ) i = (1) i = −i
4 20 3
83 20 3
i = i i = 1(1) = 1
8 4 4
4 will go into 83 20 times with 3 left.
7. This "discriminates" or tells us what type of solutions we'll have.
− b ± b − 4ac 2
ax + bx + c = 0
2
x=
2a
If we have a quadratic equation and are considering solutions
from the complex number system, using the quadratic formula,
one of three things can happen.
1. The "stuff" under the square root can be positive and we'd get
two unequal real solutions if b 2 − 4ac > 0
2. The "stuff" under the square root can be zero and we'd get one
solution (called a repeated or double root because it would factor
into two equal factors, each giving us =
if b 2 − 4acthe0same solution).
3. The "stuff" under the square root can be negative and we'd get
two complex solutions that are conjugates of each− 4ac < 0
if b 2 other.
The "stuff" under the square root is called the discriminant.