# Quotient

Quotient[m,n]

gives the integer quotient of m and n.

Quotient[m,n,d]

uses an offset d.

# Details

• Quotient is also known as the ratio or fraction.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• Quotient[m,n] gives the greatest integer no larger than m/n.
• Quotient[m,n,d] returns x such that .

# Examples

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## Basic Examples(2)

Compute the quotient of two numbers:

Plot the sequence of quotients:

## Scope(16)

### Numerical Manipulation(8)

Quotient works over integers:

Rational numbers:

Inexact real numbers:

Exact numbers:

Complex numbers:

Compute for large integers:

Compute quotients with offsets:

### Symbolic Manipulation(8)

Reduce inequalities:

Refine symbolic expressions:

Simplify expressions:

Expand Quotient in terms of Floor:

Express Quotient as a piecewise function:

Evaluate integrals:

Recurrence equation:

Generating function:

## Applications(11)

### Basic Applications(6)

Table of the quotients of the first 100 pairs of integers:

Plot the quotient of two integers:

Plot the quotient of a number of division by 2:

Plot the quotient:

Produce a 3D plot of the quotient of two functions:

Plot the quotient of two complex variables:

### Number Theory(5)

Use NestWhileList to compute Quotient for positive arguments:

Compare with the following:

Demonstrate how division works:

Count the number of positive integers less than 1000 divisible by 2 or 3, but not divisible by 6:

Direct count:

Implement the Euclidean algorithm:

Compare with:

Simplify expressions containing Quotient:

## Properties & Relations(6)

The first part of QuotientRemainder is the Quotient:

Quotient[m,n] is equivalent to Floor[m/n] for integers:

n*Quotient[m,n,d]+Mod[m,n,d] is always equal to m:

Quotient[m,n]+FractionalPart[m/n] is always equal to for positive integers:

Use PiecewiseExpand to express as a piecewise function:

Simplify expressions containing Quotient:

## Possible Issues(1)

Quotient does not automatically resolve the value:

## Interactive Examples(1)

Plot the quotient of two integers:

## Neat Examples(2)

Plot the arguments of the Fourier transform of the Quotient:

Plot the Ulam spiral of the Quotient:

Wolfram Research (1988), Quotient, Wolfram Language function, https://reference.wolfram.com/language/ref/Quotient.html (updated 2002).

#### Text

Wolfram Research (1988), Quotient, Wolfram Language function, https://reference.wolfram.com/language/ref/Quotient.html (updated 2002).

#### CMS

Wolfram Language. 1988. "Quotient." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2002. https://reference.wolfram.com/language/ref/Quotient.html.

#### APA

Wolfram Language. (1988). Quotient. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Quotient.html

#### BibTeX

@misc{reference.wolfram_2022_quotient, author="Wolfram Research", title="{Quotient}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/Quotient.html}", note=[Accessed: 04-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_quotient, organization={Wolfram Research}, title={Quotient}, year={2002}, url={https://reference.wolfram.com/language/ref/Quotient.html}, note=[Accessed: 04-June-2023 ]}