JS中数值常量的含义，数值的运算

JS采用双精度浮点数（Double-precision floating-point format， Binary64）表示数值（Number），关于计算机中浮点数的表示，可参考这里。

1 数值常量的含义

• Number.MAX_VALUE

The largest number thar can be represented in JavaScript. Equal to approximately 1.79E+308.
二进制表示：0111 1111 1110 52个1
计算公式：$(2-2^{-52})\times 2^{1023}\approx 1.8\times 10^{308}$

// true, 约等于1.79E+308
Number.MAX_VALUE === (2 - Math.pow(2, -52)) * Math.pow(2, 1023)

• Number.MIN_VALUE

The closest number to zero that can be represented in JavaScript. Equal to approximately to 5.00E-304.
二进制表示：0000 0000 0000 48个0 0001
计算公式（指数位全0）是：$2^{-1074}\approx 5^{-324}$

// true, 约等于5E-304
Number.MIN_VALUE === Math.pow(2, -52) * Math.pow(2, -1022)

• Number.MAX_SAFE_INTEGER Number.MIN_SAFE_INTEGER

The value of the largest(smallest) integer n such that n and n+1 are both exactly representable as a Number value. Equal to 9007199254740991(-9007199254740991).
二进制表示：0100 0011 0011 52个1
计算公式：$2^{-53}-1=9007199254740991$

// Number.MIN_SAFE_INTEGER is the opposite of Number.MAX_SAFE_INTEGER
// true, 9007199254740991
Number.MAX_SAFE_INTEGER === Math.pow(2, 53) - 1

• Number.EPSILON

The difference of 1 and the smallest value of number greater than 1 that is representable as Number value. Equal to approximately 2.22E-16.
二进制表示：0100 0011 0011 48个0 0001
计算公式：$2^{-52}\approx 2.22\times 10^{-16}$

// true, 约等于2.22E-16
Number.EPSILON === Math.pow(2, -52)

• Number.NaN

A value that is not a number. In equality comparations, NaN dose not equal to any value, including itself. To test whether a value is equalvalant to NaN, use isNaN function.
二进制表示：0111 1111 1111 52个不全为0

// isNaN先将参数toNumber, Number.isNaN要求参数必须是Number
isNaN === Number.isNaN()	// false

isNaN('a')		// true
isNaN('NaN')	// true
Number.isNaN('a')		// false
Number.isNaN('NaN')		// false

• Number.POSITIVE_INFINITY Number.NEGETIVE_INFINITY

A value greater than the largest number that can be represented in JavaScript. JavaScript displays it as infinity.
二进制表示：0111 1111 1111 52个不全为0

// isFinite先将参数toNumber, Number.isFinite要求参数必须是Number
isFinite === Number.isFinite	// false

isFinite('2')			// true
Number.isFinite('2')	// false

2 数值运算

• 0.1和0.2的二进制表示

// 0.0001100110011001100110011001100110011001100110011001101
// JS采用Binary64，已有精度损失，是最接近0.1的number
0.1.toString(2)

// 十进制：7205759403792794, 实际的53有效数字
let a = 0b11001100110011001100110011001100110011001100110011010
// 十进制：7205759403792793
let b = 0b11001100110011001100110011001100110011001100110011001

// JS：0.1, 计算器：0.10000000000000000555111512312578
let x = a * Math.pow(2, -56)
// JS：0.09999999999999999, 计算器：0.09999999999999999167332731531133
let y = b * Math.pow(2, -56)
// x比y更接近于0.1

// 0.001100110011001100110011001100110011001100110011001101
// 已有精度损失，是最接近0.2的number, 0.2 = 0.1 * 2
0.2.toString(2)

// 52位有效数字和0.1相同, Binary64中的52位有效数字也和0.1相同
// JS：0.2, 计算器：0.20000000000000001110223024625157
let x = a * Math.pow(2, -55)
// JS：0.19999999999999998, 计算器：0.19999999999999998334665463062265
let y = b * Math.pow(2, -55)
// x比y更接近于0.2

• 0.1 + 0.2 = ?

// 0.30000000000000004
0.1 + 0.2

// JS: 0.30000000000000004
// 计算器：0.30000000000000001665334536937735
0.10000000000000000555111512312578 + 0.20000000000000001110223024625157

// 0.0100110011001100110011001100110011001100110011001101
0.30000000000000004.toString(2)

// 十进制：5404319552844596, a = b + 1
let a = 10011001100110011001100110011001100110011001100110100
// 十进制：5404319552844595
let b = 10011001100110011001100110011001100110011001100110011

// JS：0.2, 计算器：0.30000000000000004440892098500626
let x = a * Math.pow(2, -55)
// JS：0.19999999999999998, 计算器：0.29999999999999998889776975374843
let y = b * Math.pow(2, -55)

let z = 0.30000000000000001665334536937735
x === z		// true
y === z		// false

// x比y更接近于z
x - z = 0.00000000000000002775557561562891 = 2.775557561562891E-17
z - y = 0.00000000000000002775557561562892 = 2.775557561562892E-17

3 数值的Binary64表示

• 整数的Binary64表示，MAX_SAFE_INTEGER的由来

0 ~ MAX_SAFE_INTEGER间的整数，正数的符号位是0
关键是确定指数位E和有效数字S，指数位有1023的偏移，011 1111 1111

例子1：S = 0b1.001，E = 0b1000 = $2^3$，value = 9

例子2：S = 1.11…还有50个1，E = 100…还有50个0 = $2^{52}$
$$value=(2-2^{-52})\times 2^{52}=2^{53}-1$$
小数点后的有效数字最多有52位，而E的范围足够大，能够使小数变成整数。

最后1位有效数字的权重最小，为$2^{E-52}$，E从小变大，权重也在增大，能表达的数值间隔也在增大。

• 0.1的Binary64表示

// 0.0001100110011001100110011001100110011001100110011001101
0.1.toString(2)

小数点后55位，有效数字52位，有效数字前面去掉1，后面补上0
Binary64中52位有效数字是：1001100110011001100110011001100110011001100110011010
符号位和指数位：0011 1111 1011 52位有效数字