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低位
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十进制 100
位权相加法
1 ×10²
1
0
0
0 × 10¹
0 × 10⁰
结果
100 = 1 × 10² + 0 × 10¹ + 0 ×  10⁰ 
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十进制 100
位权相加法
1 × 2³
1
0
1
0 × 2²
1 × 2¹
结果
11 = 1 × 2³ + 0 × 2² + 1 ×  2¹ + 1 × 2⁰ 
1
1 × 2⁰
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十进制 100
短除法
1
0
0
结果
10
1
0
除数
0
余数
10
1
0
10
1
0
十进制 100
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十进制 13
短除法
1
3
结果
2
除数
1
余数
6
2
0
2
1
0
二进制 1101
3
2
1
1
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zp^%L=TH(@DsO`w+@h)s9Xq4XKl6z3j#%5f6@hC(R`vMYqSnKPLSPIEFP_3WjW0Az`Mz;Ea}IRj>% zJKVvIK)Z3RL-%{y;ECu7e`&SPIhocq>smj?zd>fXvWI_qMPBJ4Twwk2tPUhPi2Wuy z;LtzmfPl5(-5=sEJGrcy-C+4t4C|3NYjx|~uwuLmyBf-XRV3o6)vdtHxl?qosg z@8w_O+dpI#<-Ke{=|zU&`EhA|3NrMcDERmH`2bfFq_*Z81}i0koMXAvBrPQ4sBgXY zjOhm!XU$XFVzgN=*ngEyj9p4SM)~L*ZBQ>`_B&;}d{h8jcH?RCzl!vT6+eMy z%d(69;o7rWFxd0sP&a`XfAuBdQWEr_zJKPT|83FN|L6IC{Q=nbR3ac&|KX=qzxLYx z>~~}T+r6ZJd-?yRgYo~opz?pEDRseeJE_#E`TjDd#Zxo-b`#KJGKY6`bllG6!~DYI zpK|E>N+-YwYko9FaM7j!mWQw*lR%jY8Z0S!9~`=S=j#z~BAU z@z + + +
二进制 011 101 001
位权相加法
0 × 2²
0
1
1
1 × 2¹
1 × 2⁰
结果
3 = × 2² + 1 ×  2¹ + 1 × 2⁰ 
1 × 2²
1
0
1
0 × 2¹
1 × 2⁰
5 = × 2² + 0 ×  2¹ + 1 × 2⁰ 
0 × 2²
0
0
1
0 × 2¹
1 × 2⁰
1 = × 2² + 0 ×  2¹ + 1 × 2⁰ 
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二进制 1110 1001
位权相加法
1 × 2³
1
1
1
1 × 2²
1 × 2¹
结果
15(E) = ×  + 1 ×   + 1 × 2¹ + 0 × 2⁰  
0
0 × 2⁰
1 × 2³
1
0
0
0 × 2²
0 × 2¹
9 = ×  + 0 ×   + 0 × 2¹ + 1 × 2⁰  
1
1 × 2⁰
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负数
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符号位
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1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
高位
低位
-0 的原码
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
符号位
最高位
真值(数据位)
机器数
-0 的反码
-0 的补码
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
溢出,丢弃
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
原码运算
0
0
0
0
0
1
1
0
6 的原码
6 + (-18)采用原码计算
-18 的原码
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
结果(原码)
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
符号位
数值位(真值)
-24
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0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
6 的原码(反码)
6 + (-18)采用反码计算
-18 的原码
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
结果(反码)
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
符号位
数值位(真值)
-18 的反码
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
结果(原码)
反码运算
-12
\ No newline at end of file diff --git a/docs/notes/01_c-basic/02_xdx/assets/33.svg b/docs/notes/01_c-basic/02_xdx/assets/33.svg index 29f26ac..0bbf08a 100644 --- a/docs/notes/01_c-basic/02_xdx/assets/33.svg +++ b/docs/notes/01_c-basic/02_xdx/assets/33.svg @@ -1,4 +1,4 @@ -
0
0
0
0
0
0
0
0
原码运算
0
0
0
0
0
1
1
0
6 的原码
6 + (-18)采用原码计算
-18 的原码
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
结果(原码)
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
符号位
数值位(真值)
-24
\ No newline at end of file +
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
18 的原码(反码)
18 + (-6)采用反码计算
-6 的原码
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
结果(反码)
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
符号位
数值位(真值)
-6 的反码
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
结果(原码)
反码运算
11
1
溢出,丢弃
\ No newline at end of file diff --git a/docs/notes/01_c-basic/02_xdx/assets/34.svg b/docs/notes/01_c-basic/02_xdx/assets/34.svg index d24337b..f31c8c0 100644 --- a/docs/notes/01_c-basic/02_xdx/assets/34.svg +++ b/docs/notes/01_c-basic/02_xdx/assets/34.svg @@ -1,4 +1,4 @@ -
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
6 的原码(反码)
6 + (-18)采用反码计算
-18 的原码
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
结果(反码)
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
符号位
数值位(真值)
-18 的反码
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
结果(原码)
反码运算
-12
\ No newline at end of file +
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
6 的原码(反码,补码)
6 + (-18)采用补码计算
-18 的原码
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
结果(补码)
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
0
符号位
数值位(真值)
-18 的反码
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
结果(反码)
补码运算
-12
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
-18 的补码
结果(原码)
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
\ No newline at end of file diff --git a/docs/notes/01_c-basic/02_xdx/assets/35.svg b/docs/notes/01_c-basic/02_xdx/assets/35.svg index 0bbf08a..2548443 100644 --- a/docs/notes/01_c-basic/02_xdx/assets/35.svg +++ b/docs/notes/01_c-basic/02_xdx/assets/35.svg @@ -1,4 +1,4 @@ -
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
18 的原码(反码)
18 + (-6)采用反码计算
-6 的原码
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
结果(反码)
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
符号位
数值位(真值)
-6 的反码
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
结果(原码)
反码运算
11
1
溢出,丢弃
\ No newline at end of file +
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
18 的原码(反码,补码)
18 + (-6)采用补码计算
-6 的原码
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
结果(补码)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
符号位
数值位(真值)
-6 的反码
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
结果(反码)
补码运算
12
1
溢出,丢弃
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
-6 的补码
结果(原码)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
6 的原码(反码,补码)
6 + (-18)采用补码计算
-18 的原码
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
结果(补码)
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
0
符号位
数值位(真值)
-18 的反码
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
结果(反码)
补码运算
-12
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
-18 的补码
结果(原码)
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
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0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
18 的原码(反码,补码)
18 + (-6)采用补码计算
-6 的原码
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
结果(补码)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
符号位
数值位(真值)
-6 的反码
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
结果(反码)
补码运算
12
1
溢出,丢弃
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
-6 的补码
结果(原码)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
\ No newline at end of file diff --git a/docs/notes/01_c-basic/02_xdx/index.md b/docs/notes/01_c-basic/02_xdx/index.md index ba88d4f..ce47fad 100644 --- a/docs/notes/01_c-basic/02_xdx/index.md +++ b/docs/notes/01_c-basic/02_xdx/index.md @@ -986,7 +986,7 @@ int main() { * 在计算机中,数据是从右往左的方式排列的;其中,最右边的是低位,最左边的是高位,即: -![](./assets/20.png) +![](./assets/20.svg) ### 3.4.2 二进制和十进制的转换 @@ -1003,13 +1003,13 @@ int main() { * 示例:十进制转十进制 -![](./assets/21.png) +![](./assets/21.svg) * 示例:二进制转十进制 -![](./assets/22.png) +![](./assets/22.svg) #### 3.4.2.2 十进制转换二进制 @@ -1025,35 +1025,35 @@ int main() { * 示例:十进制转十进制 -![](./assets/23.png) +![](./assets/23.svg) * 示例:十进制转二进制 -![](./assets/24.png) +![](./assets/24.svg) ### 3.4.3 二进制转八进制 -* 规则:每 3 位二进制就是一个八进制。 +* 规则:从右向左,每 3 位二进制就是一个八进制,不足补 0(分组转换法)。 * 示例:011 101 001 -> 351 -![](./assets/25.png) +![](./assets/25.svg) ### 3.4.4 二进制转十六进制 -* 规则:每 4 位二进制就是一个十六进制。 +* 规则:从右向左,每 4 位二进制就是一个十六进制,不足补 0(分组转换法)。 * 示例:1110 1001 -> 0xE9 -![](./assets/26.png) +![](./assets/26.svg) ## 3.5 原码、反码和补码 @@ -1061,11 +1061,19 @@ int main() { * 机器数:一个数在计算机的存储形式是二进制,我们称这些二进制数为机器数。机器数可以是有符号的,用机器数的最高位来存放符号位,0 表示正数,1 表示负数。 -![](./assets/27.png) +> [!IMPORTANT] +> +> 这里讨论的适用于`符号位`的整数,如:int 等;而不适用于`无符号位`的整数,即:unsinged int 等。 -* 真值:因为机器数带有符号位,所以机器数的形式值不等于其真实表示的值(真值),以机器数 1000 0001 为例,其真正表示的值(首位是符号位)为 -1,而形式值却是 129 ,因此将带有符号位的机器数的真正表示的值称为机器数的真值。 +![](./assets/27.svg) -![](./assets/28.png) +* 真值(数据位):因为机器数带有符号位,所以机器数的形式值不等于其真实表示的值(真值),以机器数 1000 0001 为例,其真正表示的值(首位是符号位)为 -1,而形式值却是 129 ,因此将带有符号位的机器数的真正表示的值称为机器数的真值。 + +> [!IMPORTANT] +> +> 这里讨论的适用于`符号位`的整数,如:int 等;而不适用于`无符号位`的整数,即:unsinged int 等。 + +![](./assets/28.svg) ### 3.5.2 原码 @@ -1087,7 +1095,10 @@ int main() { > [!IMPORTANT] > -> 按照原码的规则,会出现 `+0` 和 `-0` 的情况,即:`0`000 0000(+0)、`1`000 0000(-0),显然不符合实际情况;所以,计算机底层虽然存储和计算的都是二进数,但显然不是原码。 +> 总结: +> +> * ① 按照原码的规则,会出现 `+0` 和 `-0` 的情况,即:`0`000 0000(+0)、`1`000 0000(-0),显然不符合实际情况。 +> * ② 所以,计算机底层虽然存储和计算的都是二进数,但显然不是原码。 ### 3.5.3 反码 @@ -1110,7 +1121,10 @@ int main() { > [!IMPORTANT] > -> 按照反码的规则,如果是 `+0`,对应的原码是 `0`000 0000;那么,其反码还是 `0`000 0000 ;如果是 `-0`,对应的原码是 `1`000 0000,其反码是 `1`111 1111,显然不符合实际情况;所以,计算机底层虽然存储和计算的都是二进数,但显然不是反码。 +> 总结: +> +> * ① 按照反码的规则,如果是 `+0`,对应的原码是 `0`000 0000;那么,其反码还是 `0`000 0000 ;如果是 `-0`,对应的原码是 `1`000 0000,其反码是 `1`111 1111,显然不符合实际情况。 +> * ② 所以,计算机底层虽然存储和计算的都是二进数,但显然不是反码。 ### 3.5.4 补码 @@ -1130,16 +1144,18 @@ int main() { | -------- | ------------------- | ------------------- | ------------------- | | -1 | `1`000 0001 | `1`111 1110 | `1`111 1111 | -* 如果 0 ,按照 `+0` 的情况进行处理,即: +* 如果 `0` ,按照 `+0` 的情况进行处理,如下所示: -![](./assets/29.png) +![](./assets/29.svg) -* 如果 0 ,按照 `-0` 的情况进行处理,即: +* 如果 `0` ,按照 `-0` 的情况进行处理,如下所示: -![](./assets/30.png) +![](./assets/30.svg) > [!IMPORTANT] > +> 总结: +> > * ① 补码表示法解决了`原码`和`反码`存在的`两种`零(`+0` 和 `-0`)的问题,即:在补码表示法中,只有`一个`零,即 `0000 0000`。 > * ②补码使得`加法运算`和`减法运算`可以统一处理,通过将减法运算`转换`为加法运算,可以简化硬件设计,提高了运算效率。 > * ③ 计算机底层`存储`和`计算`的都是`二进数的补码`。换言之,当读取整数的时候,需要采用逆向的转换,即:将补码转换为原码。正数的原码、反码、补码都是一样的,三码合一。负数的补码转换为原码的方法就是先减去 `1` ,得到反码,再按位取反,得到原码。 @@ -1152,23 +1168,12 @@ int main() { ## 3.6 计算机底层为什么使用补码? -* 如果计算是 `2 - 2` ,那么可以转换为 `2 + (-2)`,这样计算机内部在处理`减法计算`的时候,就会将其转换为`加法计算`的形式,以简化硬件设计和提高计算效率。 +* `加法`和`减法`是计算机中最基本的运算,计算机时时刻刻都离不开它们,所以它们由硬件直接支持。为了提高加法和减法的运行效率,硬件电路必须设计得尽量简单。 +* 对于有符号位的数字来说,内存需要区分符号位和数值位:对于人类来说,很容易识别(最高位是 0 还是 1);但是,对于计算机来说,需要设计专门的电路,这无疑增加了硬件的复杂性,增加了计算时间。如果能将符号位和数值位等同起来,让它们一起参与运算,不再加以区分,这样硬件电路就可以变得非常简单。 +* 此外,加法和减法也可以合并为一种运算,即:加法运算。换言之,减去一个数就相当于加上这个数的相反数,如:`5 - 3` 相当于 `5 +(-3)`,`10 -(-9)`相当于 `10 + 9` 。 -> [!NOTE] -> -> * ① 加法和减法是计算机中最基本的运算,计算机时时刻刻都离不开它们,所以它们由硬件直接支持。为了提高加法和减法的运行效率,硬件电路必须设计得尽量简单。 -> * ② 对于有符号位的数字来说,内存需要区分符号位和数值位:对于人类来说,很容易识别(最高位是 0 还是 1);但是,对于计算机来说,需要设计专门的电路,这无疑增加了硬件的复杂性,增加了计算时间。如果能将符号位和数值位等同起来,让它们一起参与运算,不再加以区分,这样硬件电路就可以变得非常简单。 -> * ③ 此外,加法和减法也可以合并为一种运算,即:加法运算。换言之,减去一个数就相当于加上这个数的相反数,如:5 - 3 相当于 5 +(-3),10 -(-9)相当于 10 + 9 。 -> * ④ 如果能够实现上述的两个目标,那么只需要设计一种简单的、不用区分符号位和数值位的加法电路,就能同时实现加法运算和减法运算,而且非常高效。其实,这两个目标已经实现了,真正的计算机的硬件电路就是这样设计的。 -> * ⑤ 但是,简化硬件电路是有代价的,这个代价就是有符号数在存储和读取的时候都要继续转换。这也是对于有符号数的运算来说,计算机底层为什么使用`补码`的原因所在。 - -* `最高位`表示`符号位`,由于符号位的存在,如果使用`原码`来计算,就会导致`计算结果不正确`,即: - -![](./assets/31.png) - -* `补码`的设计可以巧妙的让`符号位`也参与计算,并且可以得到`正确的计算结果`,即: - -![](./assets/32.png) +* 如果能够实现上述的两个目标,那么只需要设计一种简单的、不用区分符号位和数值位的加法电路,就能同时实现加法运算和减法运算,而且非常高效。其实,这两个目标已经实现了,真正的计算机的硬件电路就是这样设计的。 +* 但是,简化硬件电路是有代价的,这个代价就是有符号数在存储和读取的时候都要继续转换。这也是对于有符号数的运算来说,计算机底层为什么使用`补码`的原因所在。 ## 3.7 补码到底是如何简化硬件电路的? @@ -1178,7 +1183,7 @@ int main() { > > 直接使用原码表示整数,让符号位也参与运算,那么对于减法来说,结果显然是不正确的。 -![](./assets/33.svg) +![](./assets/31.svg) * 于是,人们开始继续探索,不断试错,终于设计出了`反码`,如下所示: @@ -1186,7 +1191,7 @@ int main() { > > 直接使用反码表示整数,让符号位也参与运算,对于 6 +(-18)来说,结果貌似正确。 -![](./assets/34.svg) +![](./assets/32.svg) * 如果我们将`被减数`和`减数`对调一下,即:计算 `18 - 6` 的结果,也就是 `18 +(-6)`的结果,继续采用`反码`来进行运算,如下所示: @@ -1195,15 +1200,15 @@ int main() { > * ① 6 - 18,即:6+(-18),如果采用`反码`计算,结果是正确的;但是,18 - 6,即:18 +(-6),如果采用`反码`计算,结果相差 1 。 > * ② 可以推断:如果按照`反码`来计算,小数 - 大数,结果正确;而大数 - 小数,结果相差 1 。 -![](./assets/35.svg) +![](./assets/33.svg) * 对于这个相差的 `1` 必须进行纠正,但是又不能影响`小数-大数`的结果。于是,人们又绞尽脑汁设计出了`补码`,给`反码`打了一个`“补丁”`,终于把相差的 `1` 给纠正过来了。那么,`6 - 18` 按照`补码`的运算过程,如下所示: -![](./assets/36.svg) +![](./assets/34.svg) * 那么,`18 - 6` 按照`补码`的运算过程,如下所示: -![](./assets/37.svg) +![](./assets/35.svg) > [!IMPORTANT] >