Numeric Representation
Python supports different ways of representing numbers, including integer and real (floating point) numbers.
Integer Types (int)
Integers in Python can be represented in various numerical systems:
-
Decimal Notation (Base 10):
- This is the default way we represent numbers, such as
3or-89.
- This is the default way we represent numbers, such as
-
Binary Notation (Base 2):
- Represented using a prefix of
0b. It consists only of0and1. - Example:
0b1011is equivalent to11in decimal.
- Represented using a prefix of
-
Octal Notation (Base 8):
- Represented using a prefix of
0o. It uses digits from0to7. - Example:
0o12is equivalent to10in decimal.
- Represented using a prefix of
-
Hexadecimal Notation (Base 16):
- Represented using a prefix of
0x. It uses digits from0to9and lettersAtoFto represent values10to15. - Example:
0x2Bis equivalent to43in decimal.
- Represented using a prefix of
Conversion between Notations
-
To convert binary to decimal, sum each digit multiplied by
2raised to the power of its position, starting from0on the right.- Example:
0b1011in decimal is calculated as:- ( 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11 )
- Example:
-
To convert octal to decimal, use powers of
8.- Example:
0o52in decimal is calculated as:- ( 5 \times 8^1 + 2 \times 8^0 = 40 + 2 = 42 )
- Example:
-
To convert hexadecimal to decimal, use powers of
16.- Example:
0x2Bin decimal is calculated as:- ( 2 \times 16^1 + 11 \times 16^0 = 32 + 11 = 43 )
- Example:
Python can perform these conversions using built-in functions:
bin(value)returns the binary representation as a string.oct(value)returns the octal representation as a string.hex(value)returns the hexadecimal representation as a string.
To convert a string representation back to an integer, use the int() function.
- Example:
int("0b1011", 2)converts the binary string to decimal.
Integer Variables in Different Notations
When working with different notations, Python internally treats the value as an integer regardless of how it is expressed.
- Example:
All four variables have the same value,numbers.py
a = 10
b = 0b1010
c = 0o12
d = 0xA
print(a, b, c, d) # Outputs: 10 10 10 1010, but are written in different notations.
Floating-Point Numbers (float)
Floating-point numbers represent real numbers and can have a fractional part.
-
Standard Notation:
- Examples:
3.0,-89.14.
- Examples:
-
Scientific Notation:
- Represented using
eorEto indicate powers of10. - Example:
1.5e3is equivalent to ( 1.5 \times 10^3 = 1500 ).
- Represented using
Precision Issues with Floating-Point Numbers
- Floating-point numbers may have precision issues because they cannot precisely represent all real numbers due to limitations in memory representation.
- Example:
Instead ofnumbers.py
a = 0.1
b = 0.2
sum = a + b
print("The sum is", sum) # Outputs: The sum is 0.300000000000000040.3, Python returns a value close to0.3due to rounding errors in binary representation.
Addressing Precision Issues
- For cases where precision is critical (e.g., financial calculations), Python provides the
decimalmodule, which offers more accurate representation and operations. This module is beyond the scope of the video.
Practical Examples
-
Printing Variables in Different Notations:
numbers.pya = 10
b = 0b1010
c = 0o12
d = 0xA
print(bin(a), oct(b), hex(c), hex(d)) # Outputs: 0b1010 0o12 0xa 0xaThis example prints the binary, octal, and hexadecimal representations of the value
10. -
Floating-Point Precision Example:
numbers.pya = 0.1
b = 0.2
sum = a + b
print(f"The sum is approximately: {sum:.2f}") # Outputs: The sum is approximately: 0.30By formatting the output, we can display the sum as
0.30, but the underlying value still contains precision errors.