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Nonnegative

Nonnegative

Not negative. That is, greater than or equal to zero.

 

 

See also

Positive number

Key Formula

x≥0x \geq 0
Where:
  • xx = A nonnegative number (any real number that is zero or positive)

Worked Example

Problem: From the list {−5, −1, 0, 3, 7.2, −0.4}, identify all the nonnegative numbers.
Step 1: Recall that nonnegative means greater than or equal to zero.
x≥0x \geq 0
Step 2: Check each number: −5 < 0 (negative), −1 < 0 (negative), 0 = 0 (nonnegative), 3 > 0 (nonnegative), 7.2 > 0 (nonnegative), −0.4 < 0 (negative).
Step 3: Collect the numbers that passed the test.
{0,  3,  7.2}\{0,\; 3,\; 7.2\}
Answer: The nonnegative numbers in the list are 0, 3, and 7.2.

Another Example

Problem: The square root symbol √ always returns a nonnegative result. Explain why √25 is 5 and not −5.
Step 1: Both 5 and −5 satisfy the equation x² = 25, because 5² = 25 and (−5)² = 25.
52=25and(−5)2=255^2 = 25 \quad \text{and} \quad (-5)^2 = 25
Step 2: By convention, the principal square root symbol √ is defined to return the nonnegative root.
25≥0\sqrt{25} \geq 0
Step 3: Since 5 ≥ 0 and −5 < 0, the principal square root selects 5.
25=5\sqrt{25} = 5
Answer: √25 = 5 because the square root function is defined to return the nonnegative value.

Frequently Asked Questions

Is zero nonnegative?
Yes. Nonnegative means greater than or equal to zero, so zero qualifies. Zero is the boundary: it is neither positive nor negative, but it is nonnegative (and also nonpositive).
What is the difference between nonnegative and positive?
Positive means strictly greater than zero, so it excludes zero. Nonnegative means greater than or equal to zero, so it includes zero. For example, 0 is nonnegative but not positive.

Nonnegative vs. Positive

A nonnegative number satisfies x≥0x \geq 0, which includes zero. A positive number satisfies x>0x > 0, which excludes zero. Every positive number is nonnegative, but not every nonnegative number is positive — zero is the one exception. When a problem says "nonnegative," always remember that zero is a valid value.

Why It Matters

Many mathematical definitions rely on the word nonnegative. For instance, the principal square root x\sqrt{x} is defined only for nonnegative xx and always returns a nonnegative result. Lengths, areas, absolute values, and probabilities are all nonnegative quantities, so the concept appears constantly across algebra, geometry, and statistics.

Common Mistakes

Mistake: Thinking nonnegative means the same as positive and forgetting that zero counts.
Correction: Nonnegative includes zero. If a problem asks for a nonnegative integer, then 0 is a valid answer.
Mistake: Confusing nonnegative with nonpositive.
Correction: Nonnegative means ≥ 0 (zero and above). Nonpositive means ≤ 0 (zero and below). They overlap only at zero.

Related Terms