remain

The etymology is 'Nam-'[1] + '-Aji'[2] is estimated to be

If you change these words into Chinese characters , it becomes "guitar" when others.

remainder

In division, the remainder is the number that remains when the dividend (the number being divided) is divided by the divisor (the number being divided) . It is also called the remainder. If 7 is divided by 2, the quotient is 3 and the remainder is 1. If the quotient of A divided by B is C and the remainder is R, then the equation A = BC + R is established, and division can be checked with this equation. For example, in the division above, 7 = 2 × 3 + 1.

The mathematically rigorous definition of a remainder is: According to the division theorem, positive integer divisors$b$and any integer dividend$a$About,$a =bq+r$is the only integer that satisfies$q$and$0 \le r < b$exists. At this time$r$is called the remainder. In other words, in summary, the divisor is a positive integer, the dividend is irrelevant (including negative numbers), and the remainder is 0~(divisor - 1), which is the 'standard' determined in mathematics.

However, since this definition appears only in number theory , which is a university mathematics, secondary school curriculum cannot accurately define division with negative numbers and inevitably glosses over it.

Of course, there is no need to be bound by this 'standard', whether in mathematics or in everyday life. Intuitively, when dividing a negative number, the remainder is negative, i.e. the range of the remainder is$-b<r \le 0$, but this case has the disadvantage that the rest set of positive numbers and the rest set of negative numbers are different. range of the rest$-(b-1)/2 \le r \le b/2$In some cases, it is defined as a/b, in which case the quotient means 'the integer closest to a/b'. If you study number theory or algebra , you will feel that the essential issue is not 'what the remainder is' but 'whether the remainders are the same or different '. In particular, congruence , which is obtained by extracting only the remainder from an integer, is one of the core topics of number theory .

However, in reality, especially in computer instructions, how to define the rest will depend on the intended use.

If the divisor or modulo is fixed, then the term ($a_{n+k} = c_1 a_{n+k-1} + \cdots + c_k a_n$form) have the periodicity of the remainder.[3] in other words$a_n$The cycle divided by is repeated in a certain pattern.
Remainder periods of polynomials and powers can be obtained by using Euler's theorem , proto-roots , and Chinese remainder theorem appropriately . In the case of the Fibonacci sequence and the like, it is a bit difficult, but it is possible to calculate the remaining period in a similar way by using knowledge such as algebraic number theory .

In elementary mathematics, the property of the remainder is observed by observing that the last digit of a square number is repeated as 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, etc., or by observing multiples in the decimal system . You would have seen it. However, among multiple multiple identification methods, there are many that do not accurately determine the remainder.

In division of polynomials, the remainder can also be defined. polynomial$A(x)$and$B(x) \neq 0$About,$A(x) = B(x) Q(x)+R(x)$satisfies$\deg(R)<\deg(B)$polynomial$Q(x),R(x)$is the only one that exists At this time$R(x)$is defined as the remainder.

for example$x^3 + 2x+1$second$x^2$Divide by$x^3+2x+1=x^2\cdot x + (2x+1)$So, the quotient is$x$rest of it$2x+1$am. According to the rest theorem$F(x)$cast$(x-\alpha)$When divided by, the remainder is$F(\alpha)$get the

There is a point to be aware of if the division of polynomials is also followed according to the rules. If all coefficients of the quotient and remainder of the polynomial are considered as rational/real/complex numbers (i.e., considered on a sieve ), division is possible, but if only integer coefficients are insisted on, division often does not hold. example$x^2 + 1$second$2x+1$Dividing by is not possible on integer coefficients. step$B(x)$If the highest order term of is 1 or -1 (called monic), division is possible even with integer coefficients.

For polynomials with more than two variables, quotients and remainders are not conceivable.

Abstract algebra considers the Euclid domain, which expands the concept that division with remainders is possible in integers and polynomials, and considers a wider range of remainders.
It can be easily calculated using the granularity method .

Since the integer division operator (QUOTIENT) and the modulus operator (MOD) provided by Excel have different calculation methods, care should be taken when handling negative numbers.