knowledge base

A number is a power of 2 if it's greater than 0 and its binary representation has only one '1'

def is_power_of_2(n):
    return n > 0 and (n & (n - 1)) == 0

This property is why the expression n & (n - 1) == 0

• For n=21=2n = 2^1 = 2n=21=2, binary: 0010 → n−1=1n-1 = 1n−1=1, binary: 0001

• For n=22=4n = 2^2 = 4n=22=4, binary: 0100 → n−1=3n-1 = 3n−1=3, binary: 0011

• For n=23=8n = 2^3 = 8n=23=8, binary: 1000 → n−1=7n-1 = 7n−1=7, binary: 0111

数字转换

在计算机领域，单位的换算分为以 10 为底和以 2 为底的两种常用方式。以下是常见的单位及其对应的换算关系，以及在十进制和二进制中的计算方法：

1. 以 10 为底的单位（国际标准单位，SI 进制）

单位	英文单位	数值（以 10 为底）	对应的数量（英文）
KB	Kilobyte	(1 , KB = 10^3 , bytes = 1,000 , bytes)	One thousand (1 thousand)
MB	Megabyte	(1 , MB = 10^6 , bytes = 1,000,000 , bytes)	One million (1 million)
GB	Gigabyte	(1 , GB = 10^9 , bytes = 1,000,000,000 , bytes)	One billion (1 billion)
TB	Terabyte	(1 , TB = 10^{12} , bytes = 1,000,000,000,000 , bytes)	One trillion (1 trillion)
PB	Petabyte	(1 , PB = 10^{15} , bytes = 1,000,000,000,000,000 , bytes)	One quadrillion (1 quadrillion)

2. 以 2 为底的单位（计算机传统，二进制进制）

单位	英文单位	数值（以 2 为底）	对应的数量（英文）
KiB	Kibibyte	(1 , KiB = 2^{10} , bytes = 1,024 , bytes)	One thousand and twenty-four
MiB	Mebibyte	(1 , MiB = 2^{20} , bytes = 1,048,576 , bytes)	One million and forty-eight thousand five hundred seventy-six
GiB	Gibibyte	(1 , GiB = 2^{30} , bytes = 1,073,741,824 , bytes)	One billion and seventy-three million seven hundred forty-one thousand eight hundred twenty-four
TiB	Tebibyte	(1 , TiB = 2^{40} , bytes = 1,099,511,627,776 , bytes)	One trillion and ninety-nine billion five hundred eleven million six hundred twenty-seven thousand seven hundred seventy-six
PiB	Pebibyte	(1 , PiB = 2^{50} , bytes = 1,125,899,906,842,624 , bytes)	One quadrillion one hundred twenty-five trillion eight hundred ninety-nine billion nine hundred six million eight hundred forty-two thousand six hundred twenty-four

对应的英文数字表达

• (10^3 = 1,000) - Thousand

• (10^6 = 1,000,000) - Million

• (10^9 = 1,000,000,000) - Billion

• (10^{12} = 1,000,000,000,000) - Trillion

• (10^{15} = 1,000,000,000,000,000) - Quadrillion

而在二进制单位中：

• (2^{10} = 1,024)

• (2^{20} = 1,048,576)

• (2^{30} = 1,073,741,824)

• (2^{40} = 1,099,511,627,776)

• (2^{50} = 1,125,899,906,842,624)

因此，以 2 为底的单位常用于表示更精确的内存容量和数据传输，而以 10 为底的单位常用于硬盘容量、网络流量等领域。

成环问题

数组扩充成2倍遍历，就模拟了成环遍历，结果只取数组原长度。

无需真的扩充数组，直接for 循环2倍数组长度, i = i mod (len(arr))