The 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.
In Spanish
propiedad de identidad
Also called
identity property of addition, identity property of multiplication, additive identity, multiplicative identity, identity property of 0, identity property of 1
Adding 0 or multiplying by 1 changes nothing, so 0 and 1 are the identities for addition and multiplication.
Formal definition and notation
For every real numbera: a + 0 = 0 + a = a (identity property of addition) and a · 1 = 1 · a = a (identity property of multiplication). The number 0 is the only additive identity and 1 is the only multiplicative identity. An identity element e for an operation · satisfies a · e = e · a = a for all a.
Symbol
Read it as
Meaning
a + 0 = a
a plus zero equals a
identity property of addition
a · 1 = a
a times one equals a
identity property of multiplication
Key facts
The word “identity” means the number keeps its identity: it stays itself.
Subtracting 0 and dividing by 1 also leave a number unchanged, 9 − 0 = 9 and 9 ÷ 1 = 9, but 0 and 1 only work on the right side there: 0 − 9 ≠ 9.
Do not mix up the identities: a × 0 = 0 is the zero property, not an identity.
In algebra, x means 1x, and 1 · x = x by the identity property.
Try it: guess the identity rule
Interactive
The rule is hidden. Feed the machine several inputs, including a negative and a fraction such as 2.5. Every output matches its input. Which operation could do that? Then show the rule.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Identity Property examples, step by step
Example 1: Name the property
Warm-up
Which property is shown by 0 + 46 = 46?
A number is added to 0 and stays the same.
That is the identity property of addition; 0 is the additive identity.
The zero property says a × 0 = 0: the number disappears. The identity property says a × 1 = a: the number stays.
Identity Propertyvs.Inverse property
Inverses get you back to the identity: 5 + (−5) = 0 and 5 × 15 = 1. Identities are the 0 and 1 themselves.
Quick check: 3 questions
Question 1 of 3
Which equation shows the identity property of multiplication?
Show answer and explanation
Answer: C. Multiplying by 1 keeps 8 unchanged: 1 × 8 = 8. 8 × 0 = 0 is the zero property, 8 + 0 = 8 is the identity property of addition, and 8 × 8 is a square.
Question 2 of 3
What is the additive identity?
Show answer and explanation
Answer: A. Adding 0 keeps any number the same, so 0 is the additive identity. 1 is the multiplicative identity, −1 gives opposites when multiplied, and a number added to itself doubles.
Question 3 of 3
Which value of k makes 58 × kk = 58 true for any whole number k except 0?
Show answer and explanation
Answer: D.kk = 1 for every nonzero k, so the identity property keeps 58 unchanged. Choosing only 1, 5 or 8 misses that any fraction with equal numerator and denominator equals 1.
Identity Property: frequently asked questions
What is the identity property of addition and multiplication?
The identity property of addition says adding 0 does not change a number: 23 + 0 = 23. The identity property of multiplication says multiplying by 1 does not change a number: 23 × 1 = 23. Because of this, 0 is called the additive identity and 1 the multiplicative identity. Both work for every real number, including fractions and negatives.
Why is the identity property useful?
It lets you change how a number looks without changing its value. Multiplying by 44, which is 1, turns 35 into 1220 so you can add fractions. Adding +4 − 4, which is 0, is the key step in completing the square. In both cases you use an identity in disguise.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 1.OA.B.3, CCSS 3.OA.B.5
Cite this entry
Effortless Math Team. “Identity Property.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/identity-property/
Flashcard
TermIdentity Propertya + 0 = a, a × 1 = a
DefinitionAdding 0 or multiplying by 1 leaves a number unchanged: 12 + 0 = 12, 12 × 1 = 12. 0 and 1 are the identities.
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 1AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4NumberA 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 2Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5AddendAn addend is any number that is added to another number. In 8 + 5 = 13, the addends are 8 and 5, and 13 is the sum. An addition can have two or more addends, and you can add them in any order. When one addend is missing, subtract the known addend from the sum to find it.Operations · Grade 1, Grade 2, Grade 3DivisionDivision 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 5DividendThe dividend is the number being divided in a division problem. In 56 ÷ 8 = 7, the dividend is 56, the divisor is 8 and the quotient is 7. It is the total you start with before splitting it into equal groups. In a fraction such as 568, the dividend is the numerator.Operations · Grade 3, Grade 4, Grade 5Zero 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 6Completing the SquareCompleting the square is a method that rewrites x2 + bx as a perfect square by adding (b2)2: x2 + bx + (b2)2 = (x + b2)2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra · Algebra 1, Algebra 2, PrecalculusZeroZero (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.Number System · Grade 3, Grade 4, Grade 5Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6Associative PropertyThe associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.Operations · Grade 1, Grade 3, Grade 6Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8Additive 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 8