基本介紹
![可公度量](/img/5/cdb/wZwpmLyEzNycjM4MzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLzMzL0MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
作線段度量時,如果兩條線段都能用第三條線段量盡,即兩條線段都是第三條線段的整數倍,則把第三條線段稱為前兩條線段的一個 公度,這兩條線段就叫做 可公度量(或 可通約量),由於公度的 仍是一個公度,可知沒有最小的公度,但公度顯然不會超過兩條線段中的較小者,故公度中必有最大者,稱為 最大公度 。
相關分析
![可公度量](/img/9/b12/wZwpmLwYDN2YDM2MTO0AzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLxYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![可公度量](/img/3/16c/wZwpmL0YjM2kDN4cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
假設A、B 是可通約的兩個量,它們的比 是既約分數。用 輾轉相除法求m與n的最高公因數,結果一定是1。如果兩個量可通約,那么輾轉相度一定有量完的時候。這時候充當除數的那個量就是 公度(common measure);
![圖1](/img/a/26e/wZwpmL1IDO5ITN2ATN3QTN1UTM1QDN5MjM5ADMwAjMwUzLwUzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
反過來,若是輾轉相度永不停止,那么這兩個量就是叵通約的。例如等腰直角三角形ABC中,弦AB與股AC就沒有公度。這因為在AB上取AD=AC,又作AE平分∠BAC 交BC 於E,這時
![可公度量](/img/2/9b7/wZwpmLyEjNyYjN4YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzLzYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
所以
![可公度量](/img/2/03b/wZwpmLyAjM0QDO5MTM3QTN1UTM1QDN5MjM5ADMwAjMwUzLzEzL0YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/5/efd/wZwpmL2MDM5AjM3czM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3MzLwgzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
可見,也是等腰直角三角形。
當我們以AC度AB時,量一次剩下DB。以DB度AC和以DB度BC一樣,量一次剩下BE。下邊雖說可以再量,然而DB與BE互度時,工作的實質毫不減於AC與AB的互度。可見這項互度工作永遠不能休止,由此知道這是不可度的 。
相關定理
![可公度量](/img/8/b0b/wZwpmLxIjMzUzN1UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![可公度量](/img/4/457/wZwpmLxcTM2QzM0gTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL2czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/f/b63/wZwpmL1EjNxQDNzYDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLyYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
定理一 設可通約,可通約,則可通約。
![可公度量](/img/8/b0b/wZwpmLxIjMzUzN1UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![可公度量](/img/d/67e/wZwpmL2AjM1ATO2cTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3kzL2czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![可公度量](/img/c/c8f/wZwpmL3gTMycjM4QTOwADN0UTMyITNykTO0EDMwAjMwUzL0kzL3MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/d/67e/wZwpmL2AjM1ATO2cTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3kzL2czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![可公度量](/img/5/8fe/wZwpmLwIzN1gDO1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLxAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
定理二 設量可通約,那么與是可通約量,與也是可通約量。
證明:根據假設:
![可公度量](/img/e/4be/wZwpmLwQTO3QDOzgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL4AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
那么,
![可公度量](/img/d/5e3/wZwpmL1gTNxQDN0QTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzL4UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/0/f1c/wZwpmL0QDN5MzM0EjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLzMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/1/6c3/wZwpmL4MTO3YzM1EzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxMzL1IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
所以是可通約量。同理可證也是可通約量。
![可公度量](/img/8/b0b/wZwpmLxIjMzUzN1UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![可公度量](/img/7/080/wZwpmLwQTNyIjNzkjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/b/2b0/wZwpmLyUzN2ATMwgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLzUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/c/c8f/wZwpmL3gTMycjM4QTOwADN0UTMyITNykTO0EDMwAjMwUzL0kzL3MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/b/2b0/wZwpmLyUzN2ATMwgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLzUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/5/8fe/wZwpmLwIzN1gDO1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLxAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
定理三設是可通約量,而且則與,與也各是可通約量。
![可公度量](/img/e/4be/wZwpmLwQTO3QDOzgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL4AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/c/069/wZwpmL3AzN0ADN3EzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxMzL2UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/b/2b0/wZwpmLyUzN2ATMwgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLzUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/c/c8f/wZwpmL3gTMycjM4QTOwADN0UTMyITNykTO0EDMwAjMwUzL0kzL3MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![可公度量](/img/b/2b0/wZwpmLyUzN2ATMwgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLzUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![可公度量](/img/5/8fe/wZwpmLwIzN1gDO1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLxAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
證明: 所以。由此即得與是可通約量,同理可證與也是。
![可公度量](/img/8/b0b/wZwpmLxIjMzUzN1UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![可公度量](/img/7/79d/wZwpmLyMDO1UDMxQzMxADN0UTMyITNykTO0EDMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![可公度量](/img/d/8db/wZwpmL4gzN0EDNwcjNxMzM1UTM1QDN5MjM5ADMwAjMwUzL3YzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![可公度量](/img/5/8fe/wZwpmLwIzN1gDO1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLxAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
定理四 設是可通約量,是任何自然數或分數,那么,和是可通約量 。