What is an Imaginary Number?

In mathematics, an imaginary number is a number of the form a + b * i, where a, b are real numbers, and b 0, i² =-1. The term imaginary number was coined by the famous mathematician Descartes in the 17th century, because the concept at the time believed that it was a real non-existent number. It was later discovered that the real part a of the imaginary number a + b * i can correspond to the horizontal axis on the plane, and the imaginary part b corresponds to the vertical axis on the corresponding plane, so that the imaginary number a + b * i can correspond to the point (a, b) in the plane .

Trigonometric function

sin (a + bi) = sin (a) cos (bi) + sin (bi) cos (a)

= sin (a) cosh (b) + isinh (b) cos (a)

cos (a-bi) = cos (a) cos (bi) + sin (bi) sin (a)

= cos (a) cosh (b) + isinh (b) sin (a)

tan (a + bi) = sin (a + bi) / cos (a + bi)

cot (a + bi) = cos (a + bi) / sin (a + bi)

sec (a + bi) = 1 / cos (a + bi)

csc (a + bi) = 1 / sin (a + bi)

Arithmetic

(a + bi) ± (c + di) = (a ± c) + (b ± d) i

(a + bi) (c + di) = (ac-bd) + (ad + bc) i

(a + bi) / (c + di) = (ac + bd) / (c² + d²) + (bc-ad) i / (c² + d²)

r1 (isina + cosa) r2 (isinb + cosb) = r1r2 [cos (a + b) + isin (a + b)]

r1 (isina + cosa) / r2 (isinb + cosb) = r1 / r2 [cos (ab) + isin (ab)]

r (isina + cosa) ⁿ =

(isinna + cosna)

Complex conjugate

a + bi = a-bi

-(z ₁ + z ₂ ) = _ z ₁ + _z ₂

-(z ₁ -z ₂ ) = _ z ₁ -_z ₂

-(z ₁ z ₂ ) = _ z ₁ _z ₂

-(z ⁿ ) = (_ z) ⁿ

-z ₁ / z ₂ = _z ₁ / _z ₂

-z ^z = | z | ² R

power

z mz ⁿ = z ^{m + n}

z ^m / z ⁿ = z ^mn

(z ^m ) ⁿ = z ^mn

z ₁ mz ₂ ^m = (z ₁ z ₂ ) ^m

(z ^m ) ^{1 / n} = z ^{m / n}

z · z · z ... · z (n) = z ⁿ

z ₁ ⁿ = z _2- > z1 = z ₂ ^{1 / n}

ln (a + bi) = ln (a ^ 2 + b ^ 2) / 2 + i Arctan (b / a)

logai (x) = ln (x) / [i / 2 + lna]

x ^{ai + b} = x ^ai · x ^b = e ^{ialn (x)} · x ^b = x ^b [cos (alnx) + i sin (alnx).]

In mathematics, the power of an even exponent is

To trace the trajectory of an imaginary number, it is necessary to connect the occurrence process of the real number relative to it. We know that real numbers correspond to imaginary numbers, which include rational numbers and

1777 Swiss mathematician

We can

The higher power of i will continue to do the following cycles:

i ¹ = i,

i ² =-1,

i ³ =-i,

i ⁴ = 1,

i ⁵ = i,

i ⁶ =-1.

...

i ⁿ has periodicity, and the minimum positive period is 4.

i ⁴ⁿ = 1,

i ^{4n + 1} = i,

i ^{4n + 2} = -1,

i ^{4n + 3} = -i.

Due to the special operation rules of imaginary numbers, the symbol i appears

When = -1 / 2 + 3 / 2i or = -1 / 2- 3 / 2i:

² + + 1 = 0

³ = 1

Many operations on real numbers can be generalized to i, such as exponents, logarithms, and

Imaginary Original: Lawrence Mark Lesser (Armstrong Atlantic State College)

Translation: Xu Guoqiang

From ancient times to empty construction, the word Ai can now be multiplied. Everyone is shocked. Where can life really be? Tuning for a small test, I was surprised at the night light. Whether the triode is used or not, the AC circuit is always the same. Based on Jun Man asking absurd justice, negative value seeks root doubt Dou Zeng. Emotional class was used to listening to ears, and negative numbers are all involved. It's a bit complicated and confusing. But looking at the geometric triangle, flourishing wormwood shares the same meaning [].

IMAGINARY by Lawrence Mark LesserArmstrong Atlantic State University

Imaginary numbers, multiples of iEverybody wonders, "are they used in real life?" Well, try the amplifier I'm using right now-AC! You say it's absurd, this root of minus one. But the same things once were heard About the number negative one! Imaginary numbers are a bit complex, But in real mathematics, everything connects: Geometry, trig and call all see "i to i."

[] see "i to i." Means visible

a = a + i

Meaning nothing

The value of everything can be expressed as: a + bi, not just a real number. ^[4]