# 1.5 as a fraction

I know that 3÷2=1.5, but how do you write 1.5 in fraction form.

1.5 as a fraction

I know that 3÷2=1.5, but how do you write 1.5 in fraction form.

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1

You've hit a well-known snag (more precisely a pair of well-known snags...).

```
> I know that 3÷2=1.5,
```

Mathematically, yes. But there's more to this in Sage:

First snag : Sage *preparses* your input and (silently) converts them *in terms of Sage objects*. For example:

```
sage: a=3
sage: type(a)
<type 'sage.rings.integer.Integer'>
sage: a.parent()
Integer Ring
```

In this case, Sage's preparser has read the string "3" as an element of your inut, concluded that it was a numerical integer constant and created a Sage object, element of the Integer ring (i. e $\left({\mathbb Z},\,+,\, .\right)$). The (Python) `type`

of this object is the python representation of a class representing (some) sje objects, and the `parent`

of this object is the Sage representation of this class.

These are **different** from Python `int`

egers which you can get by explicit conversion:

```
sage: pa=int(3)
sage: type(pa)
<type 'int'>
```

Such objects do not belong in the hierarchy of Sage objects:

```
sage: pa.parent()
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
<ipython-input-11-7df6d5e7f9a4> in <module>()
----> 1 pa.parent()
AttributeError: 'int' object has no attribute 'parent'
```

The same is true of the preparsing of fractions:

```
> but how do you write 1.5 in fraction form.
```

Quite simply:

```
sage: b=3/2
sage: b
3/2
sage: type(b)
<type 'sage.rings.rational.Rational'>
sage: b.parent()
Rational Field
```

The latter is the Sage representation of the $\left({\mathbb Q},\,+,\, .\right)$ ring. And Sage's implementation of `/`

for the class `Integer`

somehow manages to compute the quotient of two integers in the `Rational`

field (which is, of course, also a ring). In fact, it's a bit more complicated than that, for technical reasons, but this explanation is *in essence* correct.

Below the system of Sage classes and methods and their interrelation lies the notion of (mathematical) *categories*, which justifies this implementation. But this is a large subject, which, IIUC, is usually treated at postgrad level (at least un the US, Europeans may do things a bit differently...).

See the excellent book Computational Mathematics with SageMath for a (much better) introduction...

Of course, you can attempt the division of two Python integers. Which (here's the second snag) results in:

- a (Python) integer if you're working in a Python 2-based Sage (current standard), where the division is the
*integer*division :

Example

```
sage: int(3)/int(2)
1
sage: type(int(3)/int(2))
<type 'int'>
```

- a (Python) float if you're working in a Python 3-based Sage (soon-to-be standard), where the division is the
*float*division:

Example

```
sage: type(int(3)/int(2))
<class 'float'>
```

(One may note that Python 2 objects have a `type`

, while Python 3 objects have a `class`

; neither have a `parent`

; though...).

This behaviour results from a change of heart by Guido van Rossum between Python 2 and Python 3...

0

Like this:

```
sage: QQ(1.5)
3/2
```

because `QQ`

stands for the:

```
sage: QQ
Rational Field
```

Asked: **
2019-10-28 19:48:01 -0600
**

Seen: **41 times**

Last updated: **Nov 04**

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