# 加

(由加法跳轉過嚟)

## 基本加法

### 例子

• ${\displaystyle 1+1=2}$ （一加一等於二）
• ${\displaystyle 2+2=4}$ （二加二等於四）
• ${\displaystyle 3+3=6}$ （三加三等於六）
• ${\displaystyle 1+2+3+4=10}$ （結合性）
• ${\displaystyle 2+2+2+2+2=10}$ 乘法

### 加嘅由來

「加」，根據粵語審音配詞字庫，讀「gaa1」或者「gaa3」。說文解字解釋，「加：語相增加也。從力從口。古牙切」原指講嘢假，加鹽加醋。之後演變到有增加嘅意思。喺兩漢之間，周髀算經已經有講解四則運算。

## 學習加法

### 幼兒學習

#### 加法表

BB仔一般都要用到加法表嚟學加法，利用加法表可以令佢哋容易啲記到${\displaystyle 1}$ ${\displaystyle 10}$ 嘅加法。

 ${\displaystyle 1}$ 嘅加法 ${\displaystyle 1+0=1}$  ${\displaystyle 1+1=2}$  ${\displaystyle 1+2=3}$  ${\displaystyle 1+3=4}$  ${\displaystyle 1+4=5}$  ${\displaystyle 1+5=6}$  ${\displaystyle 1+6=7}$  ${\displaystyle 1+7=8}$  ${\displaystyle 1+8=9}$  ${\displaystyle 1+9=10}$  ${\displaystyle 1+10=11}$ ${\displaystyle 2}$ 嘅加法 ${\displaystyle 2+0=2}$  ${\displaystyle 2+1=3}$  ${\displaystyle 2+2=4}$  ${\displaystyle 2+3=5}$  ${\displaystyle 2+4=6}$  ${\displaystyle 2+5=7}$  ${\displaystyle 2+6=8}$  ${\displaystyle 2+7=9}$  ${\displaystyle 2+8=10}$  ${\displaystyle 2+9=11}$  ${\displaystyle 2+10=12}$ ${\displaystyle 3}$ 嘅加法 ${\displaystyle 3+0=3}$  ${\displaystyle 3+1=4}$  ${\displaystyle 3+2=5}$  ${\displaystyle 3+3=6}$  ${\displaystyle 3+4=7}$  ${\displaystyle 3+5=8}$  ${\displaystyle 3+6=9}$  ${\displaystyle 3+7=10}$  ${\displaystyle 3+8=11}$  ${\displaystyle 3+9=12}$  ${\displaystyle 3+10=13}$ ${\displaystyle 4}$ 嘅加法 ${\displaystyle 4+0=4}$  ${\displaystyle 4+1=5}$  ${\displaystyle 4+2=6}$  ${\displaystyle 4+3=7}$  ${\displaystyle 4+4=8}$  ${\displaystyle 4+5=9}$  ${\displaystyle 4+6=10}$  ${\displaystyle 4+7=11}$  ${\displaystyle 4+8=12}$  ${\displaystyle 4+9=13}$  ${\displaystyle 4+10=14}$ ${\displaystyle 5}$ 嘅加法 ${\displaystyle 5+0=5}$  ${\displaystyle 5+1=6}$  ${\displaystyle 5+2=7}$  ${\displaystyle 5+3=8}$  ${\displaystyle 5+4=9}$  ${\displaystyle 5+5=10}$  ${\displaystyle 5+6=11}$  ${\displaystyle 5+7=12}$  ${\displaystyle 5+8=13}$  ${\displaystyle 5+9=14}$  ${\displaystyle 5+10=15}$ ${\displaystyle 6}$ 嘅加法 ${\displaystyle 6+0=6}$  ${\displaystyle 6+1=7}$  ${\displaystyle 6+2=8}$  ${\displaystyle 6+3=9}$  ${\displaystyle 6+4=10}$  ${\displaystyle 6+5=11}$  ${\displaystyle 6+6=12}$  ${\displaystyle 6+7=13}$  ${\displaystyle 6+8=14}$  ${\displaystyle 6+9=15}$  ${\displaystyle 6+10=16}$ ${\displaystyle 7}$ 嘅加法 ${\displaystyle 7+0=7}$  ${\displaystyle 7+1=8}$  ${\displaystyle 7+2=9}$  ${\displaystyle 7+3=10}$  ${\displaystyle 7+4=11}$  ${\displaystyle 7+5=12}$  ${\displaystyle 7+6=13}$  ${\displaystyle 7+7=14}$  ${\displaystyle 7+8=15}$  ${\displaystyle 7+9=16}$  ${\displaystyle 7+10=17}$ ${\displaystyle 8}$ 嘅加法 ${\displaystyle 8+0=8}$  ${\displaystyle 8+1=9}$  ${\displaystyle 8+2=10}$  ${\displaystyle 8+3=11}$  ${\displaystyle 8+4=12}$  ${\displaystyle 8+5=13}$  ${\displaystyle 8+6=14}$  ${\displaystyle 8+7=15}$  ${\displaystyle 8+8=16}$  ${\displaystyle 8+9=17}$  ${\displaystyle 8+10=18}$ ${\displaystyle 9}$ 嘅加法 ${\displaystyle 9+0=9}$  ${\displaystyle 9+1=10}$  ${\displaystyle 9+2=11}$  ${\displaystyle 9+3=12}$  ${\displaystyle 9+4=13}$  ${\displaystyle 9+5=14}$  ${\displaystyle 9+6=15}$  ${\displaystyle 9+7=16}$  ${\displaystyle 9+8=17}$  ${\displaystyle 9+9=18}$  ${\displaystyle 9+10=19}$ ${\displaystyle 10}$ 嘅加法 ${\displaystyle 10+0=10}$  ${\displaystyle 10+1=11}$  ${\displaystyle 10+2=12}$  ${\displaystyle 10+3=13}$  ${\displaystyle 10+4=14}$  ${\displaystyle 10+5=15}$  ${\displaystyle 10+6=16}$  ${\displaystyle 10+7=17}$  ${\displaystyle 10+8=18}$  ${\displaystyle 10+9=19}$  ${\displaystyle 10+10=20}$

### 十進制加法

• 可溝通性：${\displaystyle a+b=b+a}$ ，咁如果利用加法表學加法，原本要背${\displaystyle 100}$ 條加法式，而家就只需要背${\displaystyle 55}$ 條。
• 淨係加${\displaystyle 1}$ ${\displaystyle 2}$ ：淨係加${\displaystyle 1}$ 或者${\displaystyle 2}$ 對人嚟講係好自然，基本上唔使背都會識得做。
• 零：因為${\displaystyle 10}$ 係恆等元，加咗佢即係無加過嘢，對好多人嚟講都係廢。
• 乘二：一個數自己相加，變相係乘二，如果學習埋乘法，多數人會直接跳用乘法，而唔用加法。
• 好似乘二：如果兩個數係好近，例如：${\displaystyle 6+7}$ ${\displaystyle 6+5}$ ，一般會將個數乘二再減返個差，睇返個例子，就會將${\displaystyle 6\times 2+1=12+1=13}$ ${\displaystyle 6\times 2-1=12-1=11}$
• 利用${\displaystyle 10}$ ：例如：${\displaystyle 6+7}$ ，就可以將佢寫成${\displaystyle 10+3}$ ，咁就會計快啲。

#### 進位

  ¹
27
+ 59
————
86


#### 小數點加法

   4 5 . 1 0
+  0 4 . 3 4
————————————
4 9 . 4 4


### 其他進制加法

128 64 32 16 8 4 2 1 0
10000000 1000000 100000 10000 1000 100 10 1 0

${\displaystyle 1_{2}+1_{2}=10_{2}}$

${\displaystyle 101010_{2}+110101_{2}=1011111_{2}}$

## 數字加法

### 自然數

「設${\displaystyle N(S)}$ 做集${\displaystyle S}$ 嘅基數。咁兩個唔相交集${\displaystyle A,B}$ ${\displaystyle N(A)=a,N(B)=b}$ ，嘅聯合${\displaystyle a+b}$ ，就係${\displaystyle N(A\cup B)}$ 。」

${\displaystyle A\cup B}$ 就係兩個集${\displaystyle A}$ ${\displaystyle B}$ 聯合。另一個版本嘅定義，可以接受呢兩個集係有相交，咁個定義嘅處理手法就會先將兩個集相交嘅嘢整走先，再將兩個集聯合，咁就可以避免將相交嘅嘢重覆咁計。

### 整數

「對應任何整數${\displaystyle n}$ ${\displaystyle |n|}$ 係佢嘅絕對值。同時，${\displaystyle a}$ ${\displaystyle b}$ 係整數。如果${\displaystyle a}$ 或者${\displaystyle b}$ 係零，咁就當佢係恆等元（Identity）。如果${\displaystyle a}$ ${\displaystyle b}$ 都係正數，咁定義${\displaystyle a+b=|a|+|b|}$ 。如果${\displaystyle a}$ ${\displaystyle b}$ 都係負數，咁${\displaystyle a+b=-(|a|+|b|)}$ 。如果${\displaystyle a}$ ${\displaystyle b}$ 係唔同正負，咁就要定義${\displaystyle a+b}$ ${\displaystyle |a|}$ ${\displaystyle |b|}$ 之差，之後邊一個絕對值大啲，咁佢嗰個正負號，就係答案嘅正負號。」

「有兩個整數${\displaystyle a-b}$ ${\displaystyle c-d}$ ${\displaystyle a,b,c,d}$ 都係自然數。定義${\displaystyle (a-b)+(c-d)=(a+c)-(b+d)}$

### 有理數（分數）

{\displaystyle {\begin{aligned}{\frac {7}{9}}+{\frac {1}{2}}&={\frac {7\times 2}{9\times 2}}+{\frac {1\times 9}{2\times 9}}\\&={\frac {14}{18}}+{\frac {9}{18}}\\&={\frac {14+9}{18}}\\&={\frac {23}{18}}\end{aligned}}}

${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}}$

### 虛數

${\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i}$

## 高等數學應用

### 抽象代數入面嘅加法

#### 向量加法

${\displaystyle (a,b)+(c,d)=(a+c,b+d)}$

#### 矩陣加法

${\displaystyle A+B={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m2}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}+{\begin{bmatrix}2&3\\4&5\end{bmatrix}}={\begin{bmatrix}3&5\\7&9\end{bmatrix}}}$

### 將集合併

「當有兩個或以上咁多集要合併成一個，得出嗰一個集就會有之前咁多個集總和咁多樣剐。」

### 伸展距離

「當一條嘢俾人加長，咁佢個總長度就係原本咁多再加伸長咗咁多。」

## 參考

1. From Enderton (p. 138): "...select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."