Integer and Floating Point Arithmetic

Arithmetic operations with integers and floating-point numbers are foundational to programming and computational mathematics. This page explains their characteristics, operations, and differences in detail.

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Integer Arithmetic

Definition

Integers are whole numbers, positive or negative, including zero. They do not have decimal points.

Key Characteristics

1. Precision: Exact values without rounding errors.

2. Size: Limited by the storage capacity of the programming environment (e.g., 32-bit or 64-bit integers).

Operations

The following operations are commonly performed on integers:

Operation	Symbol	Example	Result
Addition	+	5 + 3	8
Subtraction	-	9 - 4	5
Multiplication	*	7 * 6	42
Division (Quotient)	//	10 // 3	3
Modulus (Remainder)	%	10 % 3	1
Exponentiation	**	2 ** 3	8

Properties

1. Division (//) truncates towards zero, returning an integer.

2. Operations are exact unless overflow occurs in environments with limited storage.

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Floating-Point Arithmetic

Floating-point numbers represent real numbers and can include fractions, expressed in the form of a base and an exponent (e.g., 3.14, 1.2e3).

Key Characteristics

1. Precision: Limited; rounding errors can occur.

2. Size: Typically represented in formats like 32-bit (single precision) or 64-bit (double precision).

3. Range: Can represent very small and very large numbers compared to integers.

Operations

The following operations are performed similarly to integers but may involve rounding:

Operation	Symbol	Example	Result
Addition	+	5.5 + 3.1	8.6
Subtraction	-	9.8 - 4.3	5.5
Multiplication	*	7.2 * 6.1	43.92
Division	/	10.0 / 3.0	3.333
Exponentiation	**	2.5 ** 3	15.625

Properties

1. Results are approximations due to rounding.

2. Special values:

1. Infinity: Result of dividing by zero (e.g., 1.0 / 0.0 -> inf).

2. NaN (Not a Number): Result of undefined operations (e.g., 0.0 / 0.0).

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Comparison of Integer and Floating-Point Arithmetic

Feature	Integer Arithmetic	Floating-Point Arithmetic
Precision	Exact	Approximate
Range	Limited	Large (with loss of precision)
Representation	Whole numbers	Fractional and whole numbers
Common Usage	Counting, indexing	Scientific calculations, measurements

Tips for Working with Numbers

1. Choose the Right Type: Use integers for exact counts and floating-point numbers for fractional or very large/small values.

2. Avoid Equality Comparisons: Avoid comparing floating-point numbers directly (e.g., a == b), as rounding errors might lead to incorrect results. Instead, use a tolerance value:
   abs(a - b) < tolerance

3. Be Aware of Overflow and Underflow:

1. Integers may overflow if the result exceeds their maximum size.

2. Floating-point numbers may underflow (too close to zero) or overflow to infinity.

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Understanding the differences and behavior of integer and floating-point arithmetic will help you write more robust and accurate programs.