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A 32-bit integer and a 32-bit floating-point number may each occupy four bytes, but they use those bits for different jobs. A conventional signed 32-bit integer represents every whole number from −2,147,483,648 to 2,147,483,647 exactly. A common IEEE 754 binary32 float reaches magnitudes around 3.4 × 1038 and can represent fractions, but it cannot represent every integer once values grow beyond about 16.8 million.
The short version: integers give exact, evenly spaced whole-number values within a bounded range; floats trade some of that exact coverage for fractions and a much wider range of magnitudes.
What “the same size” does—and does not—mean
Storage size tells you how many bits a type occupies, not how many useful values it represents or how arithmetic on those values behaves. A 32-bit integer and a 32-bit float each have 232 possible bit patterns. The types assign different meanings to those patterns: an integer uses them to encode whole numbers, while a float divides them among a sign, an exponent, and a significand. Floating-point formats also reserve patterns for values such as infinity and NaN.
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These are common conventions, not guarantees about every language’s types. Names such as int, long, and float can have implementation-dependent sizes or rules. For portable code, use a language’s fixed-width types where available and check its documentation. In C++, for example, the sizes of fundamental types are implementation-dependent within standard requirements (cppreference: fundamental types).
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How integers use their bits
An unsigned integer with n bits commonly represents values from 0 through 2n − 1. A conventional modern signed integer uses two’s complement and ranges from −2n−1 through 2n−1 − 1. So a signed 32-bit integer covers −231 through 231 − 1.
Within that range, every whole number has its own representation, with no gaps. Integer addition, subtraction, and multiplication produce exact whole-number results as long as the result remains representable and no conversion changes it. Division is different: in many languages, dividing two integers discards the fractional part or otherwise follows integer-division rules.
Do not assume overflow always wraps around. Depending on the language, type, and operation, overflow may wrap, raise an error, be undefined, or be handled another way. PostgreSQL offers a concrete database example: its four-byte integer ranges from −2,147,483,648 to 2,147,483,647 (PostgreSQL 15 numeric types).
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How floating point uses its bits
A floating-point number is conceptually represented as:
(−1)sign × significand × baseexponent
The sign records whether the value is positive or negative. The significand carries its significant digits; the exponent scales the value up or down. For common IEEE 754 binary formats, binary32 uses 32 bits: one sign bit, eight exponent bits, and 23 explicitly stored fraction bits. Normal values have an implicit leading bit, giving 24 bits of significand precision. Binary64 uses 64 bits and gives normal values 53 bits of significand precision.
The exponent is why a float can cover such a wide range. The significand is why it cannot retain every digit throughout that range. The common binary32 format reaches roughly ±3.4 × 1038, but its values become more widely spaced as their magnitude grows. IEEE 754 specifies floating-point formats and operations; a programming language still determines how its types and features map to that standard (IEEE 754-2019; IEEE floating-point overview).
Range, precision, and exactness are different
- Range is the span from the smallest to the largest magnitude a type can represent.
- Precision describes how many significant digits the representation can retain.
- Resolution is the spacing between neighboring representable values at a particular magnitude.
- Accuracy is how close a stored or computed value is to the real-world quantity it is meant to describe.
- Exactness means the stored value equals the intended mathematical value.
For integers, the spacing between adjacent values is always one, until the range ends. For floats, spacing depends on magnitude: values can be extremely close near zero, while gaps widen at large magnitudes. Thus, a float can have a much larger range than a same-width integer but much coarser resolution for large values.
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A binary32 float has about 24 bits of significand precision, often summarized as roughly seven decimal significant digits; binary64 has 53 bits, or roughly 15–16 decimal significant digits. Those are useful approximations, not promises that every operation will preserve exactly that many decimal places. Decimal round-tripping has its own guarantees: common guidance is up to nine significant digits to round-trip binary32 and 17 for binary64 (C++ digits10; C++ max_digits10).
Why floats eventually skip whole numbers
With p bits of significand precision, a binary floating-point format can represent every integer consecutively through 2p. For common formats, that means binary32 represents all consecutive integers through 224 = 16,777,216, and binary64 through 253 = 9,007,199,254,740,992.
Above those thresholds, some integers remain exactly representable, but not every integer is. For example, binary32 can represent 16,777,216 and 16,777,218, but it cannot represent every integer between them. The gap between adjacent representable values grows with magnitude. A 32-bit signed integer, by contrast, still represents every whole number in its range exactly.
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This is why “a 32-bit float can reach a bigger number” does not mean “it is a better 32-bit integer.” If the value is a count or identifier, losing the ability to distinguish neighboring integers can be a serious error even when the float’s overall range looks generous.
Why decimal fractions may not be exact
Binary floating point stores values in base two. Many familiar decimal fractions, including 0.1, do not have a finite binary expansion. A float therefore stores the nearest available binary value, and arithmetic rounds results to the destination format.
0.1 + 0.2In Python on a typical platform, this displays as 0.30000000000000004, rather than exactly 0.3. This is not a Python-specific defect: it follows from representing decimal inputs in a finite binary format. The result is deterministic under the relevant floating-point rules, but may differ slightly from the ideal real-number result. Python’s documentation explains the conversion of decimal 0.1 to the nearest representable binary fraction (Python floating-point arithmetic).
Formatting a value to two decimal places can make it print as 0.30, but that changes its display, not the underlying representation. Nor does a value with many digits necessarily describe reality accurately: a sensor reading may be noisy even if the stored number has many significant digits.
Arithmetic, comparison, and special values
Arithmetic
Integer operations are exact when their mathematical results fit and no conversion intervenes. But integer division may discard a remainder, and overflow behavior depends on the language. Floating-point operations round to the target format. They can also encounter overflow, underflow, or invalid operations; the resulting behavior depends in part on the language and runtime, even though IEEE 754 defines formats, operations, rounding, and exception conditions.
Two mathematically equivalent expressions may yield slightly different floating-point results if they perform operations in a different order, because rounding occurs along the way. Repeated additions can accumulate error, and subtracting nearly equal numbers can leave a result with few reliable significant digits.
Equality
Integer equality compares exact discrete values. For computed floating-point quantities, a == b compares the stored approximations, which may differ even when the intended mathematical results are equal. A tolerance test can be more appropriate:
abs(a - b) <= toleranceFor values that span different scales, a relative or combined absolute-and-relative tolerance may be more useful. There is no universal tolerance: select one based on the units, magnitude, expected accumulated error, and acceptable error for the application. Exact equality can still be appropriate when the values are known to be identical or when checking a carefully chosen exact value.
Special floating-point values
Common IEEE floating-point formats include values that ordinary integer types generally do not:
- Positive and negative infinity can represent results such as overflow or, in some environments, division by zero.
- NaN (“not a number”) can mark an invalid or undefined result. Under IEEE comparisons, NaN is not equal to itself, so a test such as
x == xis one way some code detects it. Database behavior can differ: PostgreSQL, for example, applies database-specific rules for NaN sorting and indexing. - Signed zero has positive and negative forms. They compare numerically equal in many contexts, though the sign can affect operations such as reciprocals.
- Subnormal values extend representation close to zero, usually with less precision than normal values.
Do not assume every language or database exposes or handles these cases identically. Consult the specific implementation’s documentation.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Conversions can lose information
Converting an integer to a float is exact only if that integer is representable in the float’s significand. A sufficiently large integer may be rounded during conversion. Converting that float back to an integer cannot recover information already lost.
Converting a float to an integer has a different risk: the fractional part may be discarded or rounded according to the language’s rules. A value outside the integer’s range, or a NaN or infinity, may cause an error or have other language-specific behavior. Do not assume a cast will safely wrap, clamp, or round unless the language explicitly says so.
In short, this path is not necessarily lossless:
large_integer → float → integerIEEE 754 covers conversions between integer and floating-point formats, but source-language conversion rules still matter (IEEE 754-2019).
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| Requirement | Usually suitable | Why |
|---|---|---|
| Counts, indexes, flags, IDs, pagination offsets | Integer | These are discrete values; integer representation avoids float rounding and skipped neighboring values. |
| Exact whole-number quantities within a known bound | Integer | Every whole number in range is represented exactly; check overflow limits. |
| Measurements, graphics coordinates, sensor values, scientific calculations | Float or double | Fractions and broad dynamic range are useful when bounded approximation is acceptable. |
| Currency or exact decimal business rules | Decimal, fixed-point, or scaled integer | These can preserve decimal-scale rules; define scale, rounding, and range explicitly. |
| Very large exact whole numbers | Arbitrary-precision integer | It can grow beyond a native fixed-width integer’s range, subject to resource costs. |
| Exact fractions or rigorous numerical bounds | Rational, interval, or specialized numeric type | Native binary floats may not provide the required exactness or error guarantees. |
For money, binary floating point is usually a poor fit when exact decimal arithmetic is required. A decimal type, fixed-point representation, or integer count of minor units such as cents can work, provided the scale and maximum value are controlled. PostgreSQL recommends exact numeric for monetary amounts and other calculations requiring exact storage and arithmetic; its real and double precision are inexact (PostgreSQL numeric types).
A practical checklist
Before choosing a numeric type, ask:
- Must each value be exact, or is a bounded approximation acceptable?
- Can it contain fractions, and must those fractions be exact decimal fractions?
- What are the smallest and largest plausible values?
- How many significant digits or how much resolution does the calculation need?
- What should happen on overflow, underflow, or invalid input?
- How will values be compared, rounded, displayed, and converted?
- Will values cross a database, file, API, or programming-language boundary?
Serialization is a separate concern from in-memory size. Text output needs enough significant digits to preserve a float through a parse-and-format round trip; binary interchange needs compatible format and conventions. A database or another language may also define casts, NaN ordering, and overflow differently from the code that created the value.
There is no universal performance winner between floats and integers. Speed depends on the processor, compiler, operation, vectorization, runtime, and workload. Choose first for correct numerical behavior, then measure performance in the actual application if it matters.
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