Zero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.
Zero splits the number line into negatives and positives and belongs to neither side. The red card is the one rule to remember: you can divide zero by a number, but never a number by zero.
Formal definition and notation
Zero is the integer 0, the additive identity of the real numbers: a + 0 = a for every numbera. It is its own opposite (−0 = 0), its absolute value is |0| = 0, and a × 0 = 0 for every a. It is even, because 0 = 2 × 0. Division by zero is undefined.
In a numeral, 0 is a placeholder: 405 has 0 tens. Without it, 405 would read 45.
In real situations 0 is a reference point, not nothing: 0 degrees, sea level, or a bank balance of 0 dollars.
Rules for zero
Rules for zero
a + 0 = a, a − 0 = a, a × 0 = 0, 0 ÷ a = 0 for a ≠ 0
a + 0 = a
zero is the additive identity: adding it changes nothing
a − a = 0
a number plus its opposite is zero, as in 5 + (−5) = 0
a × 0 = 0
any number times zero is zero
a ÷ 0
undefined, never 0 and never a
8 ÷ 0 would have to be a number q with q × 0 = 8. Every number times 0 is 0, so no such q exists. That is why division by zero is undefined.
Zero examples, step by step
Example 1: Use the rules for zero
Warm-up
Find (14 × 0) + (0 ÷ 9) + (6 − 0).
Any number times zero is zero: 14 × 0 = 0.
Zero divided by a nonzero number is zero: 0 ÷ 9 = 0.
Subtracting zero changes nothing: 6 − 0 = 6.
Add: 0 + 0 + 6 = 6.
Answer: The value is 6.
Example 2: Classify zero
Core
Is 0 even? Is it positive, negative, an integer, a whole number?
Even means a multiple of 2. 0 = 2 × 0, so 0 is even.
Positive numbers are greater than 0 and negative numbers are less than 0. Zero is not greater or less than itself, so it is neither.
The integers are …, −2, −1, 0, 1, 2, …, so 0 is an integer.
The whole numbers are 0, 1, 2, 3, …, so 0 is a whole number.
Answer: Zero is an even integer and a whole number, and it is neither positive nor negative.
Example 3: Zero as a reference point
Test-style
At 6 a.m. the temperature is −4 °F. By noon it rises 9 degrees, and by evening it falls 5 degrees. What is the evening temperature, and what does it mean?
Start at −4. Rising 9 degrees: −4 + 9 = 5.
Falling 5 degrees: 5 − 5 = 0.
The temperature is 0 °F. That is a real temperature on the scale, the boundary between above zero and below zero, not an absence of temperature.
Answer: The evening temperature is 0 degrees Fahrenheit, the point between positive and negative readings.
Common mistakes
✗ MistakeZero is a positive number.
✓ InsteadZero is even, but it is not positive or negative.
Positive means greater than 0 and negative means less than 0. Zero is the boundary between them.
✗ Mistake8 ÷ 0 = 0.
✓ Instead8 ÷ 0 is undefined. (0 ÷ 8 = 0 is fine.)
A division answer q must satisfy q × 0 = 8, and no number does. Dividing zero by 8 is fine because 0 × 8 = 0.
✗ MistakeZero is not a whole number, and it is not even.
✓ InsteadZero is a whole number, and it is even.
The whole numbers start at 0. Even numbers are multiples of 2, and 0 = 2 × 0.
✗ MistakeWriting three thousand forty-five as 345.
✓ InsteadWrite 3,045: the 0 holds the hundreds place.
A zero placeholder keeps every other digit in its correct place value.
Not to be confused with
Zerovs.Counting numbers
Counting numbers start at 1: 1, 2, 3, …. Whole numbers add 0 to that list. So 0 is a whole number but not a counting number.
The opposite of a number is the same distance from 0 on the other side: the opposite of 4 is −4. Zero is the only number that is its own opposite.
Quick check: 3 questions
Question 1 of 3
What is 6 ÷ 0?
Show answer and explanation
Answer: D. No number times 0 gives 6, so division by zero is undefined. Choice A mixes it up with 0 ÷ 6 = 0, B with 6 ÷ 1, and C with 6 ÷ 6.
Question 2 of 3
Which statement about zero is true?
Show answer and explanation
Answer: C.0 = 2 × 0, so it is even, and it is an integer. It is neither positive nor negative, and the whole numbers begin with 0.
Question 3 of 3
Which expression equals 0 for every number a?
Show answer and explanation
Answer: B. Any number times zero is zero. Choices A, C and D all equal a itself: adding 0, dividing by 1 and subtracting 0 leave a number unchanged.
Zero: frequently asked questions
Is zero an even number?
Yes. A number is even if it is a multiple of 2, and 0 = 2 × 0. Zero also fits the pattern on the number line: even and odd numbers alternate, and 0 sits between the odd numbers −1 and 1. Dividing 0 by 2 leaves no remainder, which is the usual test for an even number.
Is zero a whole number or a natural number?
Zero is a whole number: the whole numbers are 0, 1, 2, 3 and so on. It is also an integer and a rational number. Whether it counts as a natural number depends on the book; many US school textbooks start the natural (counting) numbers at 1, so they leave zero out.
Why can't you divide by zero?
Division undoes multiplication: 12 ÷ 3 = 4 because 4 × 3 = 12. For 12 ÷ 0, you would need a number that gives 12 when multiplied by 0, but every number times 0 is 0. No answer works, so the expression is undefined. 0 ÷ 0 fails too, because every number would work.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1Identity PropertyThe identity property says a number keeps its value when you add 0 or multiply by 1: a + 0 = a and a × 1 = a. So 0 is the additive identity and 1 is the multiplicative identity. For example, 15 + 0 = 15 and 15 × 1 = 15. It is how equivalent fractions work.Operations · Grade 1, Grade 3, Grade 6Zero Property of MultiplicationThe zero property of multiplication says that any number multiplied by 0 equals 0: a × 0 = 0 and 0 × a = 0. For example, 58 × 0 = 0. If even one factor in a product is 0, the whole product is 0. It explains why 5 ÷ 0 has no answer.Operations · Grade 3, Grade 4, Grade 6Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8OppositeThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Number System · Grade 6, Grade 7, Grade 8DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5Counting NumbersCounting numbers are the numbers you say when you count objects one by one: 1, 2, 3, 4, 5 and so on, without end. They start at 1, not 0, and include no fractions, decimals or negative numbers. Adding 0 to the counting numbers gives the whole numbers.Number System · Kindergarten, Grade 1, Grade 2Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1Additive InverseThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Number System · Grade 6, Grade 7, Grade 8Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6