算術、物理學和數學；或度量單位 的初步提綱
物理學和數學從計數開始1 個蘋果，2 個蘋果，等等。
因此，我們記錄中的“蘋果”是一個度量單位 ，所討論的數量是“蘋果數量”。
這演變成簡單的算術1 個蘋果加 1 個蘋果等於 2 個蘋果
30 個蘋果減去 10 個蘋果等於 20 個蘋果
引入簡寫符號 1 a p p l e + 1 a p p l e = 2 a p p l e s {\displaystyle 1\;apple+1\;apple=2\;apples} 30 a p p l e s − 10 a p p l e s = 20 a p p l e s {\displaystyle 30\;apples-10\;apples=20\;apples}
數學可以丟棄所討論的物理物件，並有幸關注抽象概念 1 + 1 = 2 {\displaystyle 1+1=2} ( 1 + 1 ) × a = 2 × a {\displaystyle (1+1)\times a=2\times a} 1 × a + 1 × a = 2 × a {\displaystyle 1\times a+1\times a=2\times a}
而在數學中，常數 a {\displaystyle a} 物理 常數，從而允許物理物件在數學方程式中表現為數學實體 1 × a p p l e + 1 × a p p l e = 2 × a p p l e {\displaystyle 1\times apple+1\times apple=2\times apple}
度量單位在數學方程式中很重要 ，因為它們代表著關鍵資訊，如果忽略這些物理常數，計算中就會出現錯誤 1 + 1 = 2 {\displaystyle 1+1=2} 1 × a p p l e + 1 × o r a n g e = 1 × a p p l e + 1 × o r a n g e {\displaystyle 1\times apple+1\times orange=1\times apple+1\times orange}
此外，在進行數學運算時必須小心 ( 3 × a p p l e s ) × ( 3 × a p p l e s ) = 9 × a p p l e s 2 {\displaystyle (3\times apples)\times (3\times apples)=9\times apples^{2}} ( 3 × a p p l e s ) × ( 3 × o r a n g e s ) = 9 × a p p l e s × o r a n g e s {\displaystyle (3\times apples)\times (3\times oranges)=9\times apples\times oranges} 時間通常以秒為單位測量
唯一一個沒有被十進位制化的度量單位（儘管這樣的系統確實存在）
距離
質量 面積通常以平方米為單位測量 10 m e t e r s × m e t e r s {\displaystyle 10\;meters\times meters} 10 s q u a r e m e t e r s {\displaystyle 10\;square\ meters} 10 m 2 {\displaystyle 10\;{\mbox{m}}^{2}}
體積通常以立方米測量 10 m e t e r s × m e t e r s × m e t e r s {\displaystyle 10\;meters\times meters\times meters} 10 c u b i c m e t e r s {\displaystyle 10\;cubic\ meters} 10 m 3 {\displaystyle 10\;{\mbox{m}}^{3}}
密度線性密度通常以千克每米測量 10 k i l o g r a m s p e r m e t e r {\displaystyle 10\;kilograms\ per\ meter} 10 kg / m {\displaystyle 10\;{\mbox{kg}}/{\mbox{m}}}
面積密度通常以千克每平方米測量 10 k i l o g r a m s p e r s q u a r e m e t e r {\displaystyle 10\;kilograms\ per\ square\ meter} 10 kg / m 2 {\displaystyle 10\;{\mbox{kg}}/{\mbox{m}}^{2}}
體積密度通常以千克每立方米測量 10 k i l o g r a m s p e r c u b i c m e t e r {\displaystyle 10\;kilograms\ per\ cubic\ meter} 10 kg / m 3 {\displaystyle 10\;{\mbox{kg}}/{\mbox{m}}^{3}} 大數 1 , 000 , 000 = 10 6 = 1 × 10 6 {\displaystyle 1,000,000=10^{6}=1\times 10^{6}} 2 , 500 , 000 = 2.5 × 10 6 {\displaystyle 2,500,000=2.5\times 10^{6}}
小數 0.001 = 10 − 3 = 1 × 10 − 3 {\displaystyle 0.001=10^{-3}=1\times 10^{-3}} 0.000234 = 2.34 × 10 − 4 {\displaystyle 0.000234=2.34\times 10^{-4}} [ 編輯 | 編輯原始碼 ] 進一步簡化書面數字 4 , 430 meters = 4.43 × 10 3 meters = 4.43 kilometers {\displaystyle 4,430{\mbox{ meters}}=4.43\times 10^{3}{\mbox{ meters}}=4.43{\mbox{ kilometers}}} 4 , 430 m = 4.43 × 10 3 m = 4.43 km {\displaystyle 4,430{\mbox{ m}}=4.43\times 10^{3}{\mbox{ m}}=4.43{\mbox{ km}}}
10 − 24 {\displaystyle 10^{-24}} = {\displaystyle =} y o c t o {\displaystyle yocto} = {\displaystyle =} y
10 − 21 {\displaystyle 10^{-21}} = {\displaystyle =} z e p t o {\displaystyle zepto} = {\displaystyle =} z
10 − 18 {\displaystyle 10^{-18}} = {\displaystyle =} a t t o {\displaystyle atto} = {\displaystyle =} a
10 − 15 {\displaystyle 10^{-15}} = {\displaystyle =} f e m t o {\displaystyle femto} = {\displaystyle =} f
10 − 12 {\displaystyle 10^{-12}} = {\displaystyle =} p i c o {\displaystyle pico} = {\displaystyle =} p
10 − 9 {\displaystyle 10^{-9}} = {\displaystyle =} n a n o {\displaystyle nano} = {\displaystyle =} n
10 − 6 {\displaystyle 10^{-6}} = {\displaystyle =} m i c r o {\displaystyle micro} = {\displaystyle =} µ
10 − 3 {\displaystyle 10^{-3}} = {\displaystyle =} m i l l i {\displaystyle milli} = {\displaystyle =} m
10 − 2 {\displaystyle 10^{-2}} = {\displaystyle =} c e n t i {\displaystyle centi} = {\displaystyle =} c
10 − 1 {\displaystyle 10^{-1}} = {\displaystyle =} d e c i {\displaystyle deci} = {\displaystyle =} d
10 1 {\displaystyle 10^{1}} = {\displaystyle =} d e k a {\displaystyle deka} = {\displaystyle =} da
10 2 {\displaystyle 10^{2}} = {\displaystyle =} h e c t o {\displaystyle hecto} = {\displaystyle =} h
10 3 {\displaystyle 10^{3}} = {\displaystyle =} k i l o {\displaystyle kilo} = {\displaystyle =} k
10 6 {\displaystyle 10^{6}} = {\displaystyle =} m e g a {\displaystyle mega} = {\displaystyle =} M
10 9 {\displaystyle 10^{9}} = {\displaystyle =} g i g a {\displaystyle giga} = {\displaystyle =} G
10 12 {\displaystyle 10^{12}} = {\displaystyle =} t e r a {\displaystyle tera} = {\displaystyle =} T
10 15 {\displaystyle 10^{15}} = {\displaystyle =} p e t a {\displaystyle peta} = {\displaystyle =} P
10 18 {\displaystyle 10^{18}} = {\displaystyle =} e x a {\displaystyle exa} = {\displaystyle =} E
10 21 {\displaystyle 10^{21}} = {\displaystyle =} z e t t a {\displaystyle zetta} = {\displaystyle =} Z
10 24 {\displaystyle 10^{24}} = {\displaystyle =} y o t t a {\displaystyle yotta} = {\displaystyle =} Y
在數學方程中，測量單位的行為類似於常數 ( 1 m + 2 m ) × 4 m = 12 m 2 {\displaystyle (1{\mbox{ m}}+2{\mbox{ m}})\times 4{\mbox{ m}}=12{\mbox{ m}}^{2}}
為了將一個單位轉換為另一個單位，我們利用一個將兩個測量值聯絡起來的方程式。 1 km = 1000 m {\displaystyle 1{\mbox{ km}}=1000{\mbox{ m}}\,}
我們可以求解並用常數 m {\displaystyle m} 1 1000 km = m {\displaystyle {\frac {1}{1000}}{\mbox{ km}}={\mbox{ m}}} [ 1 ( 1 1000 km ) + 2 ( 1 1000 km ) ] × 4 ( 1 1000 km ) = 12 ( 1 1000 km ) 2 {\displaystyle \left[1\left({\frac {1}{1000}}{\mbox{ km}}\right)+2\left({\frac {1}{1000}}{\mbox{ km}}\right)\right]\times 4\left({\frac {1}{1000}}{\mbox{ km}}\right)=12\left({\frac {1}{1000}}{\mbox{ km}}\right)^{2}} ( 1 × 10 − 3 km + 2 × 10 − 3 km ) × 4 × 10 − 3 km = 12 × 10 − 6 km 2 {\displaystyle \left(1\times 10^{-3}{\mbox{ km}}+2\times 10^{-3}{\mbox{ km}}\right)\times 4\times 10^{-3}{\mbox{ km}}=12\times 10^{-6}{\mbox{ km}}^{2}} 單位轉換的數學原理
1. In mathematical equations, units of measurement behave as constants
* (1\mbox{ m} + 2\mbox{ m})\times 4\mbox{ m} = 12\mbox{ m}^2
2. To convert from one unit of to another, we utilize an equation relating the two measurements
* 1\mbox{ km} = 1000\mbox{ m} \,
3. We can solve and substitute for the constant m
* \frac{1}{1000}\mbox{ km} = \mbox{ m}
* \left[1\left(\frac{1}{1000}\mbox{ km}\right) + 2\left(\frac{1}{1000}\mbox{ km}\right)\right]\times 4\left(\frac{1}{1000}\mbox{ km}\right) = 12 \left(\frac{1}{1000}\mbox{ km}\right)^2
* \left(1\times 10^{-3}\mbox{ km} + 2\times 10^{-3}\mbox{ km}\right)\times 4\times 10^{-3}\mbox{ km} = 12\times 10^{-6}\mbox{ km}^2
導數和小量
積分和無限量的求和 
![{\displaystyle \left[1\left({\frac {1}{1000}}{\mbox{ km}}\right)+2\left({\frac {1}{1000}}{\mbox{ km}}\right)\right]\times 4\left({\frac {1}{1000}}{\mbox{ km}}\right)=12\left({\frac {1}{1000}}{\mbox{ km}}\right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b61615739e6fd4a44b2fa8d4ba36ab7271a2614b)
