Computer Origin

• Computer’s origin is binary counting method in mathematics

What is Binary Counting Method?

Daily Decimal

Arabic numerals consist of 10 counting symbols from 0 to 9, and adopt carry system, every 10 carries one place, take 2871 as example

Where ^ represents power or exponentiation operation. Decimal digit places (thousands, hundreds, tens, etc.) are all in form of 10^n. Need special attention, any non-zero number’s 0th power is 1. In this new representation, 10 is called decimal counting method’s base.

Binary

We change base to 2, can understand binary representation. For example 110101.

Binary digit places are in form of 2^n

Other Bases

We just need to change base, can use any base to represent our numbers.

JavaScript Base Conversion

parseInt(num,8); //Octal to decimal
parseInt(num,16); //Hexadecimal to decimal 
parseInt(num).toString(8) //Decimal to octal 
parseInt(num).toString(16) //Decimal to hexadecimal 
parseInt(num,2).toString(8) //Binary to octal 
parseInt(num,2).toString(16) //Binary to hexadecimal
parseInt(num,8).toString(2) //Octal to binary 
parseInt(num,8).toString(16) //Octal to hexadecimal 
parseInt(num,16).toString(2) //Hexadecimal to binary 
parseInt(num,16).toString(8) //Hexadecimal to octal

Why Do Computers Use Binary?

Logic circuits composing computer systems usually only have two states, i.e., switch connection and disconnection.

Binary Bit Operations

Left Shift

Binary 110101 shifted left one bit, is adding a 0 at the end, so 110101 becomes 1101010

If convert 1101010 to decimal, is 106, have you noticed, 106 is exactly 2 times 53

Binary left shift one bit, is actually doubling the number.

Number Overflow

Number overflow means binary number’s digits exceed system specified digits. Currently mainstream systems support at least 32-bit integer numbers

Right Shift

Binary 110101 shifted right one bit, is removing the last bit, so 110101 becomes 11010

ps. If we convert 11010 to decimal, is 26, exactly 53 divided by 2’s integer quotient

Binary right shift one bit, is operation of dividing number by 2 and finding integer quotient

Code Example

console.log(5 << 2);  //Returns 20

console.log(1000 >>> 8);  //Returns 3

Left shift is «, why is right shift »> instead of »?

• is also right shift operation

• Highest bit in binary value is sign bit

What if number is -53? Then 32nd bit is not 0

Negative numbers:

Logical Right Shift »

Will change sign bit

Arithmetic Right Shift »

Bit “OR”

Binary “1” and “0” respectively correspond to logic “true” and “false”, can perform logic operations on bits

Logic “OR” means, among participating bits, as long as one bit is 1, final result is 1, i.e., “true”. If we align each bit of binary 110101 and 100011, perform bitwise “OR” operation, will get 110111.

Bit “AND”

Similarly, we can also perform logic “AND” operations on bits. “AND” means, among participating bits, must all be 1, then final result is 1 (true), otherwise is 0 (false). If we align each bit of binary 110101 and 100011, perform bitwise “AND” operation, will get 100001.

Bit “XOR”

Logic “XOR” differs from “OR”, it has exclusivity, meaning if participating bits are the same, final result is 0 (false), otherwise is 1 (true). So, to get 1, two participating bits must be different, this is “exclusive” meaning here. We align each bit of binary 110101 and 100011, perform bitwise “XOR” operation, can get result 10110.

javascript Bitwise AND, Bitwise OR, Bitwise XOR Operations

        var a=1;
        var b=0;
 
        //Bitwise AND &: Both operands are 1, result is 1
        a&b //Result is 0
 
        //Bitwise OR: As long as one operand is 1, result is 1
        a|b  //Result is 1
 
        //Bitwise XOR: Two numbers same, result is 0; two numbers different, result is 1.
        a^b  //Result is 1

        //Simple method: Number negation, subtract 1
        ~a //Result is -2

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