# Lesson 2

Integer and float numbers

## 1. Integer arithmetics

We already know the following operators which may be applied to numbers:
`+`

, `-`

, `*`

and `**`

. The division operator
`/`

for integers gives a floating-point real number (an object of type `float`

).
The exponentiation `**`

also returns a float when the power is negative:

print(17 / 3) # gives 5.66666666667 print(2 ** 4) # gives 16 print(2 ** -2) # gives 0.25

There's a special operation for integer division where the remainder is discarded: `//`

.
The operation that yields a remainder of such a division looks like `%`

.
Both operation always yield an object of type `int`

.

print(17 / 3) # gives 5.66666666667 print(17 // 3) # gives 5 print(17 % 3) # gives 2

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## 2. Floating-point numbers

When we read an integer value, we read a line with `input()`

and then cast a string to integer using `int()`

.
When we read a floating-point number, we need to cast the string to
float using `float()`

:

x = float(input()) print(x)

Floats with very big or very small absolute value can be written using a scientific notation.
Eg., the distance from the Earth to the Sun is 1.496·10^{11}, or `1.496e11`

in Python. The mass of one molecule of the water is 2.99·10^{-23},
or `2.99e-23`

in Python.

One can cast float objects to int objects by discarding the fraction part using the `int()`

function. This function demonstrates so called *rounding towards zero* behavior:

print(int(1.3)) # gives 1 print(int(1.7)) # gives 1 print(int(-1.3)) # gives -1 print(int(-1.7)) # gives -1

There's also a function `round()`

that performs the usual rounding:

print(round(1.3)) # gives 1 print(round(1.7)) # gives 2 print(round(-1.3)) # gives -1 print(round(-1.7)) # gives -2

Floating-point real numbers can't be represented with exact precision due to hardware limitations. This can lead to cumbersome effects. See the Python docs for the details.

print(0.1 + 0.2) # gives 0.30000000000000004

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## 3. math module

Python has many auxiliary functions for calculations with floats. They can be
found in the `math`

module.

To use this module, we need to import it first by writing the following instruction at the beginning of the program:

import math

For example, if we want to find a ceiling value for `x`

- the smallest integer not less than
`x`

- we call the appropriate function from the math module: `math.ceil(x)`

.
The syntax for calling functions from modules is always the same:
`module_name.function_name(argument_1, argument_2, ...)`

import math x = math.ceil(4.2) print(x) print(math.ceil(1 + 3.8))

There's another way to use functions from modules: to import the certain functions by naming them:

from math import ceil x = 7 / 2 y = ceil(x) print(y)

Some of the functions dealing with numbers - `int()`

,
`round()`

and `abs()`

(absolute value aka modulus) -
are built-in and don't require any imports.

All the functions of any standard Python module are documented on the official Python website. Here's the description for math module. The description of some functions is given:

Function | Description |
---|---|

Rounding | |

`floor(x)` |
Return the floor of x, the largest integer less than or equal to x. |

`ceil(x)` |
Return the ceiling of x, the smallest integer greater than or equal to x. |

Roots and logarithms | |

`sqrt(x)` |
Return the square root of x |

`log(x)` |
With one argument, return the natural logarithm of x (to base e). With two arguments, return the logarithm of x to the given base |

`e` |
The mathematical constant e = 2,71828... |

Trigonometry | |

`sin(x)` |
Return the sine of x radians |

`asin(x)` |
Return the arcsine of x, in radians |

`pi` |
The mathematical constant π = 3.1415... |

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