What is Zero: Definition, Examples, and Practice Math Problems
reviewed by Jo-ann Caballes
Updated on September 28, 2026
Zero (0) is the number that represents “nothing,” or the absence of a quantity — but in math, it is a full number with its own special properties. It is one of the integers, it is neither positive nor negative, and it sits exactly between −1 and 1 on the number line.
Zero is essential across math, from arithmetic to algebra. It holds place values in large numbers, it is the starting point of the number line, and it follows unique rules when you add, subtract, multiply, or divide with it.
What Is 0 in Math?
Zero in math does not just mean “nothing.” It is a number with unique properties that make it essential in arithmetic, algebra, and even computer science. Zero is the value we use to express that no positive or negative quantity is present.
Zero also has a long history. It was used as early as the 4th century CE by the Maya civilization to track days in their calendars. The concept of zero as a true number, with rules of its own, was developed in ancient India and the Arab world before it reached Europe. Today, zero is one of the building blocks children learn to count with.
Why Is Zero Important in Math?
Zero is important because of two special roles. First, it is the only number that leaves any other number unchanged when added or subtracted, which is called being the additive identity. Second, it acts as a placeholder, so we can tell 35 apart from 305 or 3,005 by the zeros that hold empty place values.
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Properties of Zero
Zero has its own set of arithmetic properties because it is neither positive nor negative. These rules describe what happens when you add, subtract, multiply, or divide using zero, and they hold true for every number.
Addition Property
The addition property of zero says that adding 0 to any number leaves the number unchanged. If you have a box of 5 pencils and add 0 pencils, you still have 5 pencils. In symbols, 5 + 0 = 5.
Subtraction Property
The subtraction property says that subtracting 0 from any number leaves it unchanged, so 5 − 0 = 5. It also tells us that subtracting a number from itself gives zero: if you remove all 5 pencils from a box of 5, none are left, so 5 − 5 = 0.
Multiplication Property
The multiplication property says that any number multiplied by 0 is 0. Multiplication is repeated addition, so 2 × 5 means adding 2 five times. But multiplying by 0 means adding a number zero times, which gives nothing at all — so 5 × 0 = 0.
Division Property
The division property has two parts. Dividing 0 by any non-zero number gives 0, so 0 ÷ 5 = 0. But dividing any number by 0 is impossible: mathematicians call it “undefined,” because no number multiplied by 0 could give a non-zero result. So 5 ÷ 0 is undefined.
What Does a Zero With a Line Through It Mean?
A symbol that looks like a zero with a line through it, ∅, is the symbol for the empty set — a set that has no elements at all. It is a different idea from the number 0, even though the two symbols look similar.
For example, if set X = {2, 4, 6, 8, 10} and set Y has nothing in it, then Y = ∅. Note that ∅ and 0 are not interchangeable: 0 is a number, while ∅ describes a set with no members. Separately, in computing and handwriting, a “slashed zero” is simply the digit 0 drawn with a line so it is not confused with the capital letter O.
Zero on the Number Line
On the number line, zero sits exactly between −1 and 1. Because it is neither positive nor negative, it marks the boundary between the negative numbers on the left and the positive numbers on the right.
Zero as a Placeholder
Zero works as a placeholder, showing the correct value of a number even when a place has no other digit. In the number 305, the 0 sits in the tens place, which tells us the 3 means 300 and keeps the number from collapsing into 35. Without that placeholder zero, a number like 30,005 could not be written clearly at all.
Place value can be a tricky idea for young learners, and working through it with elementary math tutors helps children see why a zero in the right spot changes everything.
Zero as a Decimal Placeholder
Zero is also a placeholder in decimals. In 4.05, the 0 holds the tenths place, so the 5 correctly means five hundredths. In 4.606, the 0 holds the hundredths place. Note one difference: a zero at the very end of a decimal, such as the 0 in 3.420, adds no value, so 3.420 is the same as 3.42.
Solved Examples on Zero
These worked examples put the properties of zero to use.
- Example 1: Solve 52 − 11 − 40 + 0 − 1. Working left to right: 52 − 11 = 41, 41 − 40 = 1, 1 + 0 = 1 (adding zero changes nothing), and 1 − 1 = 0. The answer is 0.
- Example 2: True or false — “multiplying 0 by 0 gives 1.” This is false. Any number times zero is zero, so 0 × 0 = 0.
- Example 3: Solve 17,808,332 × 50,009 × 1,999,087 × 0. Because one of the factors is 0, the entire product is 0, no matter how large the other numbers are.
- Example 4: In the number 3,805, is the 0 useless? No. The 0 holds the tens place, which keeps the 8 in the hundreds place and the 3 in the thousands place. Without it, the number would read as 385.
Number 0: Practice Problems
Zero Worksheets
Put this into practice with related Brighterly worksheets:
Frequently Asked Questions About Zero
Is Zero a Number?
Yes. Zero (0) is a real number and an integer. It represents the absence of a quantity — having none of something — but it behaves like any other number in equations. Zero is neither positive nor negative, and it serves as the starting point of the number line and the dividing line between positive and negative numbers.
Is Zero an Even Number?
Yes, zero is even. An even number is any integer that is divisible by 2 with no remainder, and 0 ÷ 2 = 0 with nothing left over. Zero also sits between two odd numbers, −1 and 1, which fits the pattern of even and odd numbers alternating along the number line.
Is Zero Positive or Negative?
Zero is neither positive nor negative. It is the only number with this property, which is why it sits exactly in the middle of the number line, between the negative numbers on the left and the positive numbers on the right. Zero marks the boundary between those two groups.
Can You Divide by Zero?
No. Dividing any number by zero is undefined in mathematics, because no number could be multiplied by zero to give a non-zero result. For example, 5 ÷ 0 has no answer. Dividing zero by another number, however, is perfectly fine: 0 ÷ 5 = 0.
Is Zero a Whole Number and an Integer?
Yes, zero is both. Whole numbers are 0, 1, 2, 3, and so on, so zero is the smallest whole number. Integers include all whole numbers and their negatives (…, −2, −1, 0, 1, 2, …), so zero is an integer too. It is the number that separates the positives from the negatives.
What Does Zero Do in Addition and Multiplication?
In addition, zero changes nothing: any number plus zero equals that same number (7 + 0 = 7), which is why zero is called the additive identity. In multiplication, zero takes over: any number times zero equals zero (7 × 0 = 0). These two rules make zero one of the most useful numbers in math.
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