數學家請進

虛數

圖片參考：http://baike.baidu.com/pic/1/11710102544318064.jpg





(1)[unreliable figure]∶虛假不實的數位
(2)[imaginary number]∶實數與虛數單位之積,亦即實部爲零的複數(如3i)

在數學裏，如果有某個數的平方是負數的話，那個數就是虛數了。所有的虛數都是複數。

“虛數”這個名詞是17世紀著名數學家笛卡爾創制，因爲當時的觀念認爲這是真實不存在的數位。後來發現虛數可對應平面上的縱軸，與對應平面上橫軸的實數同樣真實。虛數軸和實數軸構成的平面稱複平面,複平面上每一點對應著一個複數。

虛數的符號

1777年瑞士數學家歐拉開始使用符號i=√(-1)表示虛數的單位。而後人將虛數和實數有機地結合起來,寫成a+bi形式 (a、b爲實數)，稱爲複數。

虛數的歷史

由於虛數闖入數的領域時，人們對它的實際用處一無所知，在實際生活中似乎也沒有用複數來表達的量，因此，在很長的一段時間裏，人們對虛數産生過種種懷疑和誤解。笛卡爾稱“虛數”的本意是指他是假的；萊布尼茲在西元18世紀初則認爲：“虛數是美妙而奇異的神靈隱蔽所，它幾乎是既存在又不存在的兩栖物。”歐拉儘管在許多地方用了虛數，但又說一切形如√(-1)、√(-2)的數學式都是不可能有的，純屬虛幻的。

歐拉之後，挪威的一個測量學家維塞爾，提出把複數a+bi用平面上的點（a，b）來表示。後來，高斯提出了複平面的概念，終於使複數有了立足之地，也爲複數的應用開闢了道路。現在，複數一般用來表示向量（有方向的數量),這在力學、地圖學、航空學中的應用是十分廣泛的。虛數越來越顯示出其豐富的內容，真是：虛數不虛。

不表示實在數量的數詞。如下面例子中的一、三、五、九、百、千、萬等數詞都是虛數。【例】以一當十｜三五成群｜千方百計｜萬紫千紅｜九牛一毛｜龍生九子｜三月不知肉味｜。

描述虛數

虛數

原作：勞倫斯·馬克·萊瑟（阿姆斯壯大西洋州立學院） 翻譯：徐國強

虛文自古向空構，艾字如今可倍乘。
所問逢人驚詫甚，生活何處有真能？
嗟哉小試調音放，訝矣大爲掌夜燈。
三極管中知用否，交流電路肯鹹恒。
憑君漫問荒唐義，負值求根疑竇增。
情類當初聽慣耳，事關負數見折肱。
幾分繁複融學域，百計聯席悅有朋。
但看幾何三角地，蓬勃艾草意同承。




Imaginary number





圖片參考：http://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Gaussebene_Koordinatendarstellung.png/180px-Gaussebene_Koordinatendarstellung.png



圖片參考：http://en.wikipedia.org/skins-1.5/common/images/magnify-clip.png

In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory.[1]





Contents

1 Definition

1.1 Corollary
2 Geometric interpretation
3 Applications of imaginary numbers
4 History
5 See also
6 References
7 External links



Definition
Any complex number, z, can be written as


圖片參考：http://upload.wikimedia.org/math/e/7/7/e77e0e682bd347e9e7e1dec878ce8586.png
,
where
圖片參考：http://upload.wikimedia.org/math/6/0/7/607708482f5a2afc26174c21834baad3.png
is the imaginary unit, which has the defined property that:


圖片參考：http://upload.wikimedia.org/math/4/4/b/44b33da6be905320353c4c1906ac7d29.png

The number
圖片參考：http://upload.wikimedia.org/math/b/1/9/b19dfd822e2517400335d53c1556df72.png
, defined by


圖片參考：http://upload.wikimedia.org/math/4/e/0/4e0de3698dfd4a9ab0032ace23583f6f.png

is the real part of the complex number,
圖片參考：http://upload.wikimedia.org/math/e/d/f/edffd1a73cb003cdba3d07600e9aa8f2.png
, defined by


圖片參考：http://upload.wikimedia.org/math/3/2/f/32fa1bd11a7cdfce10d45ef678d142a8.png

is the imaginary part. Although Descartes originally used the term "imaginary number" to mean what is currently meant by the term "complex number", the term "imaginary number" today usually means a complex number with a real part equal to 0, that is, a number of the form i y. Zero (0) is the only number that is both real and imaginary.

Corollary


圖片參考：http://upload.wikimedia.org/math/b/0/0/b002b9e332482fd34f834c5b113e6493.png


圖片參考：http://upload.wikimedia.org/math/8/3/4/834b498313a7d9d60031a206f2be9222.png


圖片參考：http://upload.wikimedia.org/math/1/3/e/13e54d9243a424ab9ded92569cefda7c.png

…
and so on.

Geometric interpretation
Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented orthogonal to the real axis. One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. At 0 on this x-axis, draw a y-axis with "positive" direction going up; "positive" imaginary numbers then "increase" in magnitude upwards, and "negative" imaginary numbers "decrease" in magnitude downwards. This vertical axis is often called the "imaginary axis" and is denoted
圖片參考：http://upload.wikimedia.org/math/f/9/d/f9d3b767e53c2f6939003eef8c0690ed.png
or simply Im.
In this model, multiplication by − 1 corresponds to a rotation of 180 degrees about the origin. Multiplication by i corresponds to a 90-degree rotation in the "positive" direction (i.e. counter-clockwise), and the equation i2 = − 1 is interpreted as saying that if we apply 2 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that − i also solves the equation x2 = − 1 — see imaginary unit.
In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with a changing current, traditionally denoted by i.

Applications of imaginary numbers
...

抄wiki而已, 有啥意思