In this context, the words "ratio" and "rational" are related; a rational number is any number that can be written as a quotient of whole numbers. Rational numbers can be written in decimal form, but not all decimal numbers are rational. A number is rational only if you can write it as a quotient of whole numbers. Quotients with zero in the denominator are also irrational.
The number q is called the quotientwhile r is called the remainder.
See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder. The remainder, as defined above, is called the least positive remainder or simply the remainder. In this case, s is called the least absolute remainder.
A unique remainder can be obtained in this case by some convention such as always taking the positive value of s. In the division of 43 by 5 we have: In the division of 42 by 5 we have: In these examples, the negative least absolute remainder is obtained from the least positive remainder by subtracting 5, which is d.
This holds in general. When dividing by d, either both remainders are positive and therefore equal, or they have opposite signs. For floating-point numbers[ edit ] When a and d are floating-point numberswith d non-zero, a can be divided by d without remainder, with the quotient being another floating-point number.
If the quotient is constrained to being an integer, however, the concept of remainder is still necessary. Extending the definition of remainder for floating-point numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition, see modulo operation.
In programming languages[ edit ] Main article: Modulo operation While there are no difficulties inherent in the definitions, there are implementation issues that arise when negative numbers are involved in calculating remainders.
Different programming languages have adopted different conventions: PerlPython only modern versionsand Common Lisp choose the remainder with the same sign as the divisor d.
Euclidean division of polynomials Euclidean division of polynomials is very similar to Euclidean division of integers and leads to polynomial remainders. Its existence is based on the following theorem: Given two univariate polynomials a x and b x with b x not the zero polynomial defined over a field in particular, the reals or complex numbersthere exist two polynomials q x the quotient and r x the remainder which satisfy:The Greatest Common Divisor (GCD) of two whole numbers, also called the Greatest Common Factor (GCF) and the Highest Common Factor (HCF), is the largest whole number that's a divisor (factor) of both of them.
For instance, the largest number that divides into both 20 and 16 is 4. (Both 16 and Algebra -> Rational-functions-> SOLUTION: Show that the number is rational by writing it as a quotient of two integers. Any help with this equation would be GREATLY appreciated.
Any help with this equation would be GREATLY appreciated. You can put this solution on YOUR website! How do you write the repeating decimal as a quotient of integers?
N = Now since there are 2 digits that repeat multiply both sides by or N = Now write the first equation underneath the second equation, linng up the decimal points: N = How to Write a Rational Number as the Quotient of Two Integers By Chris Deziel; Updated April 23, You can write the ratio between the two numbers 5 and 7 as or as 5/7.
Here are a few familiar quotients of integers: 1/2 2/3 3/4 7/10 The quotient of two integers is seldom an integer. Euclidean division of polynomials is very similar to Euclidean division of integers and leads to polynomial remainders.
Its existence is based on the following theorem: Given two univariate polynomials a(x) and b(x) (with b(x) not the zero polynomial) defined over a field (in particular, the reals or complex numbers), there exist two polynomials q(x) (the quotient) and r(x) (the remainder.