Identity Property Of Addition And Multiplication, Explained
The short version
- The identity property of addition says that adding 0 to any number leaves the number unchanged: a + 0 = a. That’s why 0 is called the additive identity.
- The identity property of multiplication says multiplying any number by 1 leaves it unchanged: a × 1 = a. That makes 1 the multiplicative identity.
- Subtraction and division don’t have a true identity. 9 – 0 = 9, but 0 – 9 = -9, so 0 only works on one side.
- Don’t mix it up with the zero property of multiplication (a × 0 = 0). Multiplying by 0 changes the number. Adding 0 doesn’t.
- The idea matters later: equivalent fractions, unit conversions and a lot of algebra are “multiplying by 1” or “adding 0” in disguise.
The identity property of addition says that when you add zero to any number, the answer is that same number. In symbols, a + 0 = a and 0 + a = a. So 58 + 0 = 58, and 0 + 3.7 = 3.7. The number keeps its “identity.”
There’s a matching rule for multiplication, using 1 instead of 0. Below we’ll cover both, explain why subtraction and division are trickier than many worksheets let on, show how these properties get used in fractions and algebra, and finish with practice problems and answers.
The identity property of addition
For any number a:
a + 0 = a
0 + a = a
It works for every kind of number:
14 + 0 = 14
0 + 250 = 250
-6 + 0 = -6
3/4 + 0 = 3/4
x + 0 = x
Zero is called the additive identity because it’s the only number that does this. Add 1, or 0.001, or anything else, and the number changes. If you think of addition as moving along a number line, adding 0 means taking zero steps. You stay put.
The identity property of multiplication
For any number a:
a × 1 = a
1 × a = a
Examples:
9 × 1 = 9
1 × 42 = 42
-8 × 1 = -8
2/5 × 1 = 2/5
One is the multiplicative identity. The picture: 9 × 1 means one group of 9, which is just 9.
| Addition | Multiplication | |
|---|---|---|
| Identity number | 0 | 1 |
| Name | Additive identity | Multiplicative identity |
| Rule | a + 0 = a | a × 1 = a |
| Works in either order? | Yes | Yes |
What about subtraction and division?
Many sources say 0 is the identity for subtraction and 1 is the identity for division. That’s only half true, and it’s worth knowing why.
Subtracting 0 does leave a number alone: 9 – 0 = 9. But flip the order and it breaks: 0 – 9 = -9. A true identity has to work on both sides, the way 0 + 9 and 9 + 0 both give 9.
Division has the same issue. 12 ÷ 1 = 12, but 1 ÷ 12 = 1/12.
So in elementary school, “subtracting 0 or dividing by 1 doesn’t change a number” is a fine thing to know. Just don’t call them identity properties on a middle school test. Formally, the identity properties belong to addition and multiplication.
Identity property vs the other number properties
Kids usually learn several properties at once, and the names blur together. Here’s how they compare:
| Property | Addition | Multiplication |
|---|---|---|
| Identity | a + 0 = a | a × 1 = a |
| Zero property | (none) | a × 0 = 0 |
| Inverse | a + (-a) = 0 | a × (1/a) = 1, for a not 0 |
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
A quick way to sort them: identity keeps the number the same. The zero property turns it into 0. Inverse brings it back to the identity. Commutative swaps the order. Associative moves the parentheses.
Where the identity property shows up later
It looks too simple to matter, but “multiply by 1” and “add 0” are two of the most-used moves in all of school math. The trick is that the 1 or the 0 is usually disguised.
Equivalent fractions: multiplying by 1
To rewrite 2/3 with a denominator of 12, multiply by 4/4, which equals 1:
2/3 × 4/4 = 8/12
The value didn’t change, because 4/4 is just 1. That’s the identity property of multiplication, and it’s the whole reason equivalent fractions are equal.
Unit conversions: multiplying by 1
12 inches and 1 foot are the same length, so 12 in / 1 ft equals 1. To convert 3 feet:
3 ft × (12 in / 1 ft) = 36 in
Algebra: adding 0
Students completing the square add a number and subtract it right back, which is adding 0:
x² + 6x = x² + 6x + 9 – 9 = (x + 3)² – 9
Solving x + 0 = 23 is also just the identity property of addition: x = 23.
Common mistakes with the identity property
- Mixing up a × 0 and a + 0. The most common one by far. 7 + 0 = 7 is identity. 7 × 0 = 0 is the zero property. Kids who blend them write 7 × 0 = 7.
- Calling 0 – a the identity. 0 – 5 is -5, not 5.
- Thinking 1 is the identity for addition. Adding 1 changes the number. Only 0 works for addition, and only 1 works for multiplication.
- Not spotting a disguised 1. Kids who don’t see that 5/5 = 1 think multiplying by 5/5 changes a fraction, and then equivalent fractions feel like a rule to memorize.
Practice problems
- 58 + 0 = ?
- Fill in the blank: 0 + ___ = 14
- Which property is shown: 9 × 1 = 9?
- Which property is shown: 6 × 0 = 0?
- Solve for x: x + 0 = 23
- Use the identity property of multiplication to rewrite 3/5 with a denominator of 20.
- True or false: 0 – 9 = 9.
- Simplify: (4y + 0) × 1
Answers
- 58 (identity property of addition).
- 14.
- Identity property of multiplication.
- Zero property of multiplication. It’s not the identity property, since the answer changed to 0.
- x = 23.
- 3/5 × 4/4 = 12/20.
- False. 0 – 9 = -9. That’s why 0 isn’t a true identity for subtraction.
- 4y. Adding 0 leaves 4y, and multiplying by 1 leaves 4y.
If your kid is stuck on number properties
Property names are vocabulary, and vocabulary is easy to forget. What matters more is whether your child understands what 0 and 1 actually do. A kid who writes 7 × 0 = 7 usually doesn’t have a clear picture of multiplication as groups (seven groups of zero is nothing). Fixing that picture fixes the property. Our guide to math skills by grade level shows when multiplication meaning should be solid.
The payoff comes later. Students who never really got “multiplying by a form of 1” often hit a wall with equivalent fractions, and then with algebra. We explain that chain in why algebra struggles usually trace back to fractions. If you suspect older gaps, a math assessment will show you where to start.
Frequently asked questions
What is the identity property of addition in simple words?
Adding zero to a number doesn’t change it. 25 + 0 is still 25.
Why is it called the identity property?
Because the number keeps its identity. After you add 0 or multiply by 1, you get back exactly the number you started with.
What is an example of the identity property of addition?
0 + 17 = 17. So is -4 + 0 = -4, and y + 0 = y in algebra.
What’s the difference between the identity property and the zero property?
The identity property of addition (a + 0 = a) keeps the number the same. The zero property of multiplication (a × 0 = 0) turns any number into 0.
Is there an identity property of subtraction?
Not a true one. a – 0 = a, but 0 – a = -a, so 0 only works on the right side. That’s why textbooks at higher levels list identity properties for addition and multiplication only.
What grade learns the identity property?
Kids use it in first and second grade when adding 0, and multiplying by 1 comes with multiplication in third grade. The property names often get formal attention again in sixth grade and pre-algebra.
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