算术、物理学和数学；或度量单位 的初步提纲
物理学和数学从计数开始1 个苹果，2 个苹果，等等。
因此，我们记录中的“苹果”是一个度量单位 ，所讨论的数量是“苹果数量”。
这演变成简单的算术1 个苹果加 1 个苹果等于 2 个苹果
30 个苹果减去 10 个苹果等于 20 个苹果
引入简写符号 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}
数学可以丢弃所讨论的物理对象，并有幸关注抽象概念 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}
而在数学中，常数 a {\displaystyle a} 物理 常数，从而允许物理对象在数学方程式中表现为数学实体 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}
度量单位在数学方程式中很重要 ，因为它们代表着关键信息，如果忽略这些物理常数，计算中就会出现错误 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，在进行数学运算时必须小心 ( 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} 时间通常以秒为单位测量
唯一一个没有被十进制化的度量单位（尽管这样的系统确实存在）
距离
质量 面积通常以平方米为单位测量 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}}
体积通常以立方米测量 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}}
密度线性密度通常以千克每米测量 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}}}
面积密度通常以千克每平方米测量 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}}
体积密度通常以千克每立方米测量 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}} 大数 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}}
小数 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}} [ 编辑 | 编辑源代码 ] 进一步简化书面数字 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
在数学方程中，测量单位的行为类似于常数 ( 1 m + 2 m ) × 4 m = 12 m 2 {\displaystyle (1{\mbox{ m}}+2{\mbox{ m}})\times 4{\mbox{ m}}=12{\mbox{ m}}^{2}}
为了将一个单位转换为另一个单位，我们利用一个将两个测量值联系起来的方程式。 1 km = 1000 m {\displaystyle 1{\mbox{ km}}=1000{\mbox{ m}}\,}
我们可以求解并用常数 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}} 单位转换的数学原理
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
导数和小量
积分和无限量的求和 
![{\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)
