Deep understanding of remainder/modulo operation

Daily programming often comes into contact with the remainder/modulus operation, so how is the remainder/modulo operation defined in the computer?

P1: Four "rounding" methods

mathematical rounding

1.1 Round to 0

parseInt() 注意接受参数是string;所以调用该方法时存在类型转换

parseInt(3.141592654) // 3

1.2 Round down

Math.floor() 不四舍五入

Math.floor(1.5555) // 1

1.3 Round up

Math.ceil() 向上取整

Math.ceil(1.5555) // 2
Math.ceil(1.1555) // 2

1.4 Rounding off

Math.round()  四舍五入取整

Math.round(1.5555) //2
Math.round(1.4999) //1

P2: Use formulas to understand "modulo"

Modulo concept

If a and d are two natural numbers and d is non-zero, it can be proved that there are two unique integers q and r, satisfying  a = q*d + r 且 0 ≤ r < d. where q is called the quotient and r is called the remainder.

 Operator --- remainder operation %

 

P3: Is "remainder" the same as "modulo"?

Rounding

Remainder: Make the quotient as far as possible, and round to 0.

Modulo: Make the quotient as far as possible, and round towards -∞.

In C, % is essentially taking the remainder.
% in Python is actually a modulo.

Understand the chain:

For any number greater than 0, perform 0-direction rounding and -∞ rounding, and the rounding direction is the same. So taking the modulus is equivalent to taking the remainder

For any number less than 0, perform 0-direction rounding and -∞ rounding, and the rounding direction is opposite. Therefore, taking the modulus is not equivalent to taking the remainder

The quotient obtained by dividing with the symbol data must be a positive number (positive number vs positive integer), that is, greater than 0! Therefore, when rounding the quotient, taking the modulus is equivalent to taking the remainder.

P4: Summary

• There are many rounding methods for floating point numbers (or division of integers), such as rounding to 0, rounding to (positive, negative) infinity, rounding and rounding...
• If a and d are two natural numbers, and d is non-zero, it can be proved that there are two unique integers q and r, satisfying a = q*d + r, q is an integer, and 0 ≤ |r|< |d|. where q is called the quotient and r is called the remainder
• In different languages, the results of "modulo" are different for the same calculation expression. We can call them the positive and negative remainders respectively
• The size of the specific remainder r essentially depends on the quotient q. And the quotient depends on the rounding rules when calculating the division.
• Remainder vs Modulo: Remainder is to get the quotient as far as possible and round towards 0. The modulus is to get the quotient as far as possible, rounding towards -∞.
• If the two data involved in the remainder have the same sign, taking the modulus is equivalent to taking the remainder
• If the two data symbols involved in the remainder are different, in C language (or other languages ​​that use rounding towards 0 such as: C++, Java), the symbol of the remainder is the same as the dividend. (Because it is rounded towards 0)