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Positive Number

Positive Number

A real number greater than zero. Zero itself is not positive.

 

 

See also

Negative number, nonnegative number

Key Formula

x>0x > 0
Where:
  • xx = A real number; if this inequality holds, then x is positive

Example

Problem: Classify each number as positive, negative, or neither: 7, −3, 0, 0.01, −½.
Step 1: Check whether 7 is greater than, less than, or equal to zero.
7>0⇒positive7 > 0 \quad \Rightarrow \quad \text{positive}
Step 2: Check −3.
−3<0⇒negative-3 < 0 \quad \Rightarrow \quad \text{negative}
Step 3: Check 0. Zero is neither greater than nor less than itself.
0=0⇒neither positive nor negative0 = 0 \quad \Rightarrow \quad \text{neither positive nor negative}
Step 4: Check 0.01. Even though it is very small, it lies to the right of zero on the number line.
0.01>0⇒positive0.01 > 0 \quad \Rightarrow \quad \text{positive}
Step 5: Check −½.
−12<0⇒negative-\tfrac{1}{2} < 0 \quad \Rightarrow \quad \text{negative}
Answer: 7 and 0.01 are positive; −3 and −½ are negative; 0 is neither.

Frequently Asked Questions

Is zero a positive number?
No. Zero is neither positive nor negative. It sits at the boundary between positive and negative numbers on the number line. If a problem requires zero to be included with the positives, mathematicians use the term "nonnegative" (meaning zero or greater).
Can fractions and decimals be positive?
Yes. Any fraction or decimal greater than zero is positive. For instance, 0.001, 3/7, and 2.5 are all positive numbers. The concept applies to every type of real number, not just whole numbers.

Positive vs. Nonnegative

A positive number satisfies x>0x > 0, which excludes zero. A nonnegative number satisfies x≥0x \geq 0, which includes zero. So every positive number is nonnegative, but zero is nonnegative without being positive. This distinction matters when stating domains, ranges, or conditions in algebra and calculus.

Why It Matters

Positive numbers appear everywhere: lengths, distances, prices, and temperatures above zero are all positive quantities. Many formulas require positive inputs — for example, you can only take the square root of a nonnegative number in the real number system, and logarithms are defined only for positive numbers. Recognizing whether a value is positive, negative, or zero also determines the direction of an inequality when you multiply or divide both sides.

Common Mistakes

Mistake: Treating zero as a positive number.
Correction: Zero is neither positive nor negative. If you need to include zero, use the term "nonnegative" (x≥0x \geq 0) instead of "positive" (x>0x > 0).
Mistake: Assuming "positive" means only positive integers (1, 2, 3, …).
Correction: Positive numbers include all real numbers greater than zero: integers, fractions, decimals, and irrational numbers like 2\sqrt{2} and π\pi.

Related Terms

  • Negative Number — Real numbers less than zero
  • Nonnegative — Zero or any positive number
  • Zero — The boundary between positive and negative
  • Real Numbers — The set containing all positive, negative, and zero
  • Number Line — Visual representation showing positive numbers right of zero
  • Absolute Value — Always returns a nonnegative result
  • Sign — Indicates whether a number is positive, negative, or zero