Example 1
- a = 3, b = 7
7
What is max(3, 7)? 7.
Maximum of two numbers calculator
Enter two real numbers a and b. The result is the larger one: max(a, b). For 3 and 7 the value is 7. For −2 and 5 the value is 5. When a and b are equal, the result equals that common value. This page does not compute argmax or the maximum of a list longer than two arguments.
Result
Enter a and b.
Two fields, one result. Min, max, and range with a third number live on minimum and maximum of numbers. List range — on range (max − min).
How it works
a, b3, 7
|a−b|4
max7
The maximum of two real numbers is the larger one with respect to the usual order on ℝ. Piecewise: max(a, b) = a when a ≥ b; otherwise max(a, b) = b. The calculator uses the numbers you enter in fields a and b and computes max(a, b) — the same result as the classical definition.
An equivalent absolute-value identity is max(a, b) = (a + b + |a − b|) / 2. The midpoint of the segment is (a+b)/2, and half the distance is |a−b|/2; their sum reaches the larger endpoint. When a ≥ b, |a−b| = a−b and the expression simplifies to a; when b > a it simplifies to b. The minimum has the twin formula min(a, b) = (a + b − |a − b|) / 2. Hence max + min = a + b and max − min = |a − b|.
When a = b both branches give the same value; |a−b| = 0. Field order does not change the result (commutativity). Negatives and zero compare as on the line: −2 and 5 give 5; 0 and −3 give 0; −8 and −3 give −3. Worked examples: 3 and 7 → 7; −2 and 5 → 5; 4 and 4 → 4; 2.5 and 1 → 2.5.
max returns a value, not an index. argmax would indicate which field wins; a tie does not pick a unique index. Min, max, and range of three numbers live on a neighbouring page; list range on another. An empty field leaves the result blank. A comma and a period are equivalent.
Mathematical model: maximum with respect to the order on ℝ and the identity max = (a+b+|a−b|)/2 (standard in real analysis / algebra textbooks). Not a 0–100 scale or a custom “index” — only the larger of two numbers.
Formula
max(a, b) = a when a ≥ b; otherwise max(a, b) = b.
Equivalently: max(a, b) = (a + b + |a − b|) / 2. Minimum: min(a, b) = (a + b − |a − b|) / 2.
Computation: max(a, b). Model: maximum with respect to the order on ℝ + absolute-value identity.
Theory, identity, and order on ℝ
Definition
The maximum of two real numbers a and b, written max(a, b), is the larger of them: max(a, b) = a when a ≥ b, and b when b ≥ a. An equivalent branch-free algebraic identity is max(a, b) = (a + b + |a − b|)/2. Dually min(a, b) = (a + b − |a − b|)/2. The calculator uses max(a, b) and shows a check of the absolute-value identity.
The definition is standard in analysis and in Concrete Mathematics (Knuth et al.); MathWorld (Maximum) gives the same |·| formula. In programming max is a built-in library function.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Historical and conceptual context
Comparing two magnitudes is prior to the full min/max lattice on sets. In optimization, statistics (sample extrema), and graphics (clipping), max is everyday. The |a − b| identity lets one write max in languages that have abs but awkward branching — useful in proofs and in smoothed automatic differentiation.
Conceptually max of two numbers does not sort lists or compute percentiles. A separate min–max–range calculator handles two or three values together with the range.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Formal properties
Domain: a, b ∈ ℝ. Codomain: ℝ. max(a, a) = a; max(a, b) = max(b, a); max(a, max(b, c)) = max(max(a, b), c). max(a, b) ≥ a and max(a, b) ≥ b.
Links: max(a, b) + min(a, b) = a + b; max(a, b) − min(a, b) = |a − b|. Positive homogeneity: max(λa, λb) = λ max(a, b) for λ ≥ 0; for λ < 0 the inequality flips toward min.
The formula (a + b + |a − b|)/2 follows by checking the cases a ≥ b and a < b.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Full derivation
Step 1. If a ≥ b, then |a − b| = a − b, so (a + b + a − b)/2 = a = max(a, b).
Step 2. If a < b, then |a − b| = b − a, so (a + b + b − a)/2 = b = max(a, b).
Step 3. In both cases the formula holds. Analogously min(a, b) = (a + b − |a − b|)/2.
Step 4. max selects by comparison; the result must match the identity in floating-point arithmetic up to the representation of |·|.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Special cases
Examples: max(3, 7) = 7; max(−2, 5) = 5; max(4, 4) = 4; max(0, −3) = 0; max(−8, −3) = −3. When a = b, |a − b| = 0 and the formula yields a.
NaN in computer arithmetic propagates through max in a browser-dependent way; this calculator requires finite numeric inputs.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Numerical and computational notes
The comparison a ≥ b is exact for finite numbers other than NaN. The abs identity may overflow a + b for huge |a|, |b| even when max is finite — hence library max is safer; the panel still shows both routes on typical data.
Comma and period are equivalent. The axis diagram marks both values and the maximum.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Related concepts and authorities
Related: min–max–range (list of 2–3 numbers), floor, square of a sum. The median of three is a separate model.
Authorities: Concrete Mathematics; MathWorld — Maximum; Wikipedia — Maxima and minima; ECMAScript Math.max. Educational tool.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
How to use
- Enter number a (e.g. 3) and number b (e.g. 7).
- Click Calculate or wait for the live result. 3 and 7 give 7. −2 and 5 give 5.
- Check the panel: min, |a−b|, the (a+b+|a−b|)/2 check, and the number line with the highlighted maximum.
- For equal values (4 and 4) the result is 4 — a tie.
- Min/max/range with a third number lives on the neighbouring page — here only max of two fields.
Symbol glossary
max(a, b) = (a+b+|a−b|)/2. For 3 and 7 the maximum is 7. The panel also shows min and |a−b|.
- max
- The larger of a and b. 3 and 7 → 7. −8 and −3 → −3. 4 and 4 → 4.
- a, b
- Two real numbers from the form fields. Order does not change max.
- |a−b|
- Distance on the line. In the identity: max = (a+b+|a−b|)/2.
- min
- The smaller of a and b: (a+b−|a−b|)/2. Shown in the panel as context, not as the primary result.
- argmax
- The winning index, not the value. Here the primary result is always the max value.
Examples
Example 2
- a = −2, b = 5
5
What is max(−2, 5)? 5.
Example 3
- a = 4, b = 4
4
What is max(4, 4)? 4.
Example 4
- a = 0, b = −3
0
What is max(0, −3)? 0.
Example 5
- a = 2.5, b = 1
2.5
What is max(2.5, 1)? 2.5.
Example 6
- a = −8, b = −3
−3
What is max(−8, −3)? −3.
Example 7
- a = 7, b = 3
7
What is max(7, 3)? 7 (same as 3 and 7).
Example 8
- a = −1, b = −1
−1
What is max(−1, −1)? −1.
Related calculators
Common questions
What does this calculator compute?
It returns the larger of two real numbers a and b, that is max(a, b). For a = 3 and b = 7 the result is 7. One result — not a minimum and not a three-number range.
What is the piecewise definition of max(a, b)?
If a ≥ b, then max(a, b) = a; otherwise max(a, b) = b. When a = b both branches give the same value. On the number line the larger number lies farther to the right.
Where does the identity (a+b+|a−b|)/2 come from?
The midpoint of the segment [a, b] is (a+b)/2, and half the distance is |a−b|/2. Their sum reaches the larger endpoint. When a ≥ b, |a−b| = a−b, so (a+b+a−b)/2 = a. When b > a, |a−b| = b−a and the result is b. This is a classical real-analysis identity.
How does the minimum follow from this?
min(a, b) = (a+b−|a−b|)/2. Also max(a,b)+min(a,b)=a+b and max(a,b)−min(a,b)=|a−b|. The result panel shows the minimum and |a−b| beside the primary maximum.
What happens when a = b?
The result equals a (and also b). |a−b| = 0, so the absolute-value identity returns a as well. Example: 4 and 4 give 4.
Does the order of fields a and b change the result?
No. max is commutative: max(a,b)=max(b,a). Example 7: a = 7, b = 3 also gives 7 — the same as 3 and 7.
How do negatives and zero work?
Comparison is the usual order on ℝ. −2 and 5 give 5; 0 and −3 give 0; −8 and −3 give −3 (because −3 > −8). Zero is a number, not an empty field.
What happens with an empty field?
An empty a or empty b leaves the result blank. The calculator does not substitute the other number for a missing entry.
How does max differ from argmax?
max(a,b) returns the value (the larger number). argmax would indicate which field wins (a or b), and a tie does not pick a unique index. The panel shows a winner label or a tie, but the primary result is always the max value.
Where do I compute min, max, and range of three numbers?
On the minimum and maximum of numbers page (a, b, and optional c). Range from a list lives on the separate range calculator. Here only max of two fields.
Do a comma and a period work the same way?
Yes. 2.5 and 2,5 are the same number. For a = 2.5 and b = 1 the result is 2.5.
Do the currency and unit settings in the page header affect the result?
No. The result depends only on a and b. The currency and unit settings in the page header do not affect this result.
How do I calculate max(a, b)?
Pick the larger of the two numbers, or use max(a, b) = (a + b + |a − b|)/2. For a = 3 and b = 7: (10 + 4)/2 = 7.
What is max(−2, 5)?
5. The positive number lies farther right on the line than −2. That is Example 2 on this page.
What is max(−8, −3)?
−3, because −3 > −8. That is Example 6 on this page.
Does the calculator show which field won (argmax)?
No. The result is the maximum value, not a field index. On a tie both numbers equal the result.
Does the page compute the maximum of three or more numbers?
No. Here only two arguments. Min, max, and range of three numbers and list range live on neighbouring cards.
Knowledge sources
The definition, |·| identity, and links to min below come from recognized sources.
- Wikipedia — Maximum and minimum
- ProofWiki — Maximum in terms of absolute value
- MathWorld — Maximum
- Math Stack Exchange — max/min via a+b and |a−b|
- Rudin, W., Principles of Mathematical Analysis (orders and bounds in ℝ; classic analysis text).
Page Updated November 2, 2026.