Python - Numbers

---

Most of the times you work with numbers in whatever you do. Obviously, any computer application deals with numbers. Hence, programming languages, Python included, have built-in support to store and process numeric data.

In this chapter, we shall learn about properties of Python number types in detail.

Three number types, integers (int), floating point numbers (float) and complex numbers, are built into the Python interpreter. Python also has a bult-in Boolean data type called bool. It can be treated as a sub-type of int type, since its two possible values True and False represent the integers 1 and 0 respectively.

Python − Integers

In Python, any number without the provision to store a fractional part is an integer. (Note that if the fractional part in a number is 0, it doesn't mean that it is an integer. For example a number 10.0 is not an integer, it is a float with 0 fractional part whose numeric value is 10.) An integer can be zero, positive or a negative whole number. For example, 1234, 0, -55.

There are three ways to form an integer object. With literal representation, any expression evaluating to an integer, and using int() function.

Literal is a notation used to represent a constant directly in the source code. For example −

>>> a =10

However, look at the following assignment of the integer variable c.

a=10
b=20
c=a+b
print ("a:", a, "type:", type(a))

It will produce the following output −

a: 10 type: <class 'int'>

Here, c is indeed an integer variable, but the expression a+b is evaluated first, and its value is indirectly assigned to c.

The third method of forming an integer object is with the return value of int() function. It converts a floating point number or a string in an integer.

>>> a=int(10.5)
>>> b=int("100")

You can represent an integer as a binary, octal or Hexa-decimal number. However, internally the object is stored as an integer.

Binary Numbers

A number consisting of only the binary digits (1 and 0) and prefixed with 0b is a binary number. If you assign a binary number to a variable, it still is an int variable.

A represent an integer in binary form, store it directly as a literal, or use int() function, in which the base is set to 2

a=0b101
print ("a:",a, "type:",type(a))
b=int("0b101011",2)
print ("b:",b, "type:",type(b))

It will produce the following output −

a: 5 type: <class 'int'>
b: 43 type: <class 'int'>

There is also a bin() function in Python. It returns a binary string equivalent of an integer.

a=43
b=bin(a)
print ("Integer:",a, "Binary equivalent:",b)

It will produce the following output −

Integer: 43 Binary equivalent: 0b101011

Octal Numbers

An octal number is made up of digits 0 to 7 only. In order to specify that the integer uses octal notation, it needs to be prefixed by 0o (lowercase O) or 0O (uppercase O). A literal representation of octal number is as follows −

a=0O107
print (a, type(a))

It will produce the following output −

71 <class 'int'>

Note that the object is internally stored as integer. Decimal equivalent of octal number 107 is 71.

Since octal number system has 8 symbols (0 to 7), its base is 7. Hence, while using int() function to covert an octal string to integer, you need to set the base argument to 8.

a=int('20',8)
print (a, type(a))

It will produce the following output −

16 <class 'int'>

Decimal equivalent of octal 30 is 16.

In the following code, two int objects are obtained from octal notations and their addition is performed.

a=0O56
print ("a:",a, "type:",type(a))
b=int("0O31",8)
print ("b:",b, "type:",type(b))
c=a+b
print ("addition:", c)

It will produce the following output −

a: 46 type: <class 'int'>
b: 25 type: <class 'int'>
addition: 71

To obtain the octal string for an integer, use oct() function.

a=oct(71)
print (a, type(a))

Hexa-decimal Numbers

As the name suggests, there are 16 symbols in the Hexadecimal number system. They are 0-9 and A to F. The first 10 digits are same as decimal digits. The alphabets A, B, C, D, E and F are equivalents of 11, 12, 13, 14, 15, and 16 respectively. Upper or lower cases may be used for these letter symbols.

For the literal representation of an integer in Hexadecimal notation, prefix it by 0x or 0X.

a=0XA2
print (a, type(a))

It will produce the following output −

162 <class 'int'>

To convert a Hexadecimal string to integer, set the base to 16 in the int() function.

a=int('0X1e', 16)
print (a, type(a))

Try out the following code snippet. It takes a Hexadecimal string, and returns the integer.

num_string = "A1"
number = int(num_string, 16)
print ("Hexadecimal:", num_string, "Integer:",number)

It will produce the following output −

Hexadecimal: A1 Integer: 161

However, if the string contains any symbol apart from the Hexadecimal symbol chart (for example X001), it raises following error −

Traceback (most recent call last):
  File "C:\Python311\var1.py", line 4, in <module>
    number = int(num_string, 16)
ValueError: invalid literal for int() with base 16: 'X001'

Python's standard library has hex() function, with which you can obtain a hexadecimal equivalent of an integer.

a=hex(161)
print (a, type(a))

It will produce the following output −

0xa1 <class 'str'>

Though an integer can be represented as binary or octal or hexadecimal, internally it is still integer. So, when performing arithmetic operation, the representation doesn't matter.

a=10 #decimal
b=0b10 #binary
c=0O10 #octal
d=0XA #Hexadecimal
e=a+b+c+d
print ("addition:", e)

It will produce the following output −

addition: 30

Python − Floating Point Numbers

A floating point number has an integer part and a fractional part, separated by a decimal point symbol (.). By default, the number is positive, prefix a dash (-) symbol for a negative number.

A floating point number is an object of Python's float class. To store a float object, you may use a literal notation, use the value of an arithmetic expression, or use the return value of float() function.

Using literal is the most direct way. Just assign a number with fractional part to a variable. Each of the following statements declares a float object.

>>> a=9.99
>>> b=0.999
>>> c=-9.99
>>> d=-0.999

In Python, there is no restriction on how many digits after the decimal point can a floating point number have. However, to shorten the representation, the E or e symbol is used. E stands for Ten raised to. For example, E4 is 10 raised to 4 (or 4^th power of 10), e-3 is 10 raised to -3.

In scientific notation, number has a coefficient and exponent part. The coefficient should be a float greater than or equal to 1 but less than 10. Hence, 1.23E+3, 9.9E-5, and 1E10 are the examples of floats with scientific notation.

>>> a=1E10
>>> a
10000000000.0
>>> b=9.90E-5
>>> b
9.9e-05
>>> 1.23E3
1230.0

The second approach of forming a float object is indirect, using the result of an expression. Here, the quotient of two floats is assigned to a variable, which refers to a float object.

a=10.33
b=2.66
c=a/b
print ("c:", c, "type", type(c))

It will produce the following output −

c: 3.8834586466165413 type <class 'float'>

Python's float() function returns a float object, parsing a number or a string if it has the appropriate contents. If no arguments are given in the parenthesis, it returns 0.0, and for an int argument, fractional part with 0 is added.

>>> a=float()
>>> a
0.0
>>> a=float(10)
>>> a
10.0

Even if the integer is expressed in binary, octal or hexadecimal, the float() function returns a float with fractional part as 0.

a=float(0b10)
b=float(0O10)
c=float(0xA)
print (a,b,c, sep=",")

It will produce the following output −

2.0,8.0,10.0

The float() function retrieves a floating point number out of a string that encloses a float, either in standard decimal point format, or having scientific notation.

a=float("-123.54")
b=float("1.23E04")
print ("a=",a,"b=",b)

It will produce the following output −

a= -123.54 b= 12300.0

In mathematics, infinity is an abstract concept. Physically, infinitely large number can never be stored in any amount of memory. For most of the computer hardware configurations, however, a very large number with 400th power of 10 is represented by Inf. If you use "Infinity" as argument for float() function, it returns Inf.

a=1.00E400
print (a, type(a))
a=float("Infinity")
print (a, type(a))

It will produce the following output −

inf <class 'float'>
inf <class 'float'>

One more such entity is Nan (stands for Not a Number). It represents any value that is undefined or not representable.

>>> a=float('Nan')
>>> a
Nan

Python − Complex Numbers

In this section, we shall know in detail about Complex data type in Python. Complex numbers find their applications in mathematical equations and laws in electromagnetism, electronics, optics, and quantum theory. Fourier transforms use complex numbers. They are Used in calculations with wavefunctions, designing filters, signal integrity in digital electronics, radio astronomy, etc.

A complex number consists of a real part and an imaginary part, separated by either "+" or "−". The real part can be any floating point (or itself a complex number) number. The imaginary part is also a float/complex, but multiplied by an imaginary number.

In mathematics, an imaginary number "i" is defined as the square root of -1 ($\sqrt{−1}$). Therefore, a complex number is represented as "x+yi", where x is the real part, and "y" is the coefficient of imaginary part.

Quite often, the symbol "j" is used instead of "I" for the imaginary number, to avoid confusion with its usage as current in theory of electricity. Python also uses "j" as the imaginary number. Hence, "x+yj" is the representation of complex number in Python.

Like int or float data type, a complex object can be formed with literal representation or using complex() function. All the following statements form a complex object.

>>> a=5+6j
>>> a
(5+6j)
>>> type(a)
<class 'complex'>
>>> a=2.25-1.2J
>>> a
(2.25-1.2j)
>>> type(a)
<class 'complex'>
>>> a=1.01E-2+2.2e3j
>>> a
(0.0101+2200j)
>>> type(a)
<class 'complex'>

Note that the real part as well as the coefficient of imaginary part have to be floats, and they may be expressed in standard decimal point notation or scientific notation.

Python's complex() function helps in forming an object of complex type. The function receives arguments for real and imaginary part, and returns the complex number.

There are two versions of complex() function, with two arguments and with one argument. Use of complex() with two arguments is straightforward. It uses first argument as real part and second as coefficient of imaginary part.

a=complex(5.3,6)
b=complex(1.01E-2, 2.2E3)
print ("a:", a, "type:", type(a))
print ("b:", b, "type:", type(b))

It will produce the following output −

a: (5.3+6j) type: <class 'complex'>
b: (0.0101+2200j) type: <class 'complex'>

In the above example, we have used x and y as float parameters. They can even be of complex data type.

a=complex(1+2j, 2-3j)
print (a, type(a))

It will produce the following output −

(4+4j) <class 'complex'>

Surprised by the above example? Put "x" as 1+2j and "y" as 2-3j. Try to perform manual computation of "x+yj" and you'll come to know.

complex(1+2j, 2-3j)
=(1+2j)+(2-3j)*j
=1+2j +2j+3
=4+4j

If you use only one numeric argument for complex() function, it treats it as the value of real part; and imaginary part is set to 0.

a=complex(5.3)
print ("a:", a, "type:", type(a))

It will produce the following output −

a: (5.3+0j) type: <class 'complex'>

The complex() function can also parse a string into a complex number if its only argument is a string having complex number representation.

In the following snippet, user is asked to input a complex number. It is used as argument. Since Python reads the input as a string, the function extracts the complex object from it.

a= "5.5+2.3j"
b=complex(a)
print ("Complex number:", b)

It will produce the following output −

Complex number: (5.5+2.3j)

Python's built-in complex class has two attributes real and imag − they return the real and coefficient of imaginary part from the object.

a=5+6j
print ("Real part:", a.real, "Coefficient of Imaginary part:", a.imag)

It will produce the following output −

Real part: 5.0 Coefficient of Imaginary part: 6.0

The complex class also defines a conjugate() method. It returns another complex number with the sign of imaginary component reversed. For example, conjugate of x+yj is x-yj.

>>> a=5-2.2j
>>> a.conjugate()
(5+2.2j)