Maximum and Minimum of Two Numbers Using Absolute Value (With Proof)

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Maximum and Minimum of Two Numbers Using Absolute Value

The maximum and minimum of two real numbers a and b can be expressed using the absolute value function. These formulas are very useful in mathematical analysis and proofs.

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1. Formula for Maximum

max(a, b) = (a + b + |a − b|) / 2

Proof

Case 1: a ≥ b

a − b ≥ 0 ⇒ |a − b| = a − b

(a + b + |a − b|) / 2
= (a + b + (a − b)) / 2
= a

Hence, max(a, b) = a.

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Case 2: b > a

a − b < 0 ⇒ |a − b| = b − a

(a + b + |a − b|) / 2
= (a + b + (b − a)) / 2
= b

Hence, max(a, b) = b.

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2. Formula for Minimum

min(a, b) = (a + b − |a − b|) / 2

Proof

Case 1: a ≤ b

a − b ≤ 0 ⇒ |a − b| = b − a

(a + b − |a − b|) / 2
= (a + b − (b − a)) / 2
= a

Hence, min(a, b) = a.

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Case 2: b < a

a − b > 0 ⇒ |a − b| = a − b

(a + b − |a − b|) / 2
= (a + b − (a − b)) / 2
= b

Hence, min(a, b) = b.

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Important Identity

max(a, b) + min(a, b) = a + b

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