The Identity Property of Zero in Addition - First Grade Mathematics

Zero is a very quiet number, but it has one of the most powerful properties in all of mathematics. Have you ever wondered what happens when you add zero to a number? Absolutely nothing changes! If you have four shiny pennies in your pocket and nobody gives you any more pennies, you still have exactly four pennies. In mathematics, this rule is known as the Identity Property of Zero in Addition. In this chapter, you will discover why zero leaves every number unchanged, explore how to model zero with concrete objects, and learn to solve addition equations with zero in a flash.

What Does Zero Mean in Addition?

Zero represents an empty set, or the quantity of "none."

Adding Zero to 4:
Plate A:   (Cookie) (Cookie) (Cookie) (Cookie)   -> 4 Cookies
Plate B:   (Empty Plate! No cookies at all!)     -> 0 Cookies
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Put both plates together on the table:
Total:     (Cookie) (Cookie) (Cookie) (Cookie)   -> 4 Cookies!

Equation: 4 + 0 = 4

The Identity Rule

In everyday language, your "identity" is who you are; it stays the same no matter what clothes you wear. In mathematics, the "identity" of a number is its value. When you add 0 to any number, that number keeps its identity. It does not grow larger, and it does not shrink; it stays exactly the same!

The Identity Rule:
Any Number + 0 = The Same Number!
0 + Any Number = The Same Number!

Adding Zero in Equations

Let us look at equations with zero where zero is the second addend, or where zero is the first addend.

Equations with Zero:
3 + 0 = 3          0 + 3 = 3
7 + 0 = 7          0 + 7 = 7
9 + 0 = 9          0 + 9 = 9
0 + 0 = 0          10 + 0 = 10

Zero on the Left or Right

Whether zero is added at the end (5 + 0) or at the beginning (0 + 5), the sum is always 5. Adding zero means adding nothing to the group.

Modeling Zero on a Number Path

When you add on a number path, each number tells you how many steps to hop forward.

Adding Zero on a Number Path:
Start on 6:      [ 6 ]
Instruction:     + 0  (Take ZERO hops forward!)
Where do you land? You stay right on [ 6 ]!
6 + 0 = 6

The Zero Hop

Taking zero hops means your feet stay completely still! Because you took no steps, your position does not change.

Practice Exercises

Exercise 1: Solve each of these addition equations with zero: 8 + 0 = __ 0 + 5 = __ 0 + 0 = __ 10 + 0 = __

Exercise 2: You have an aquarium with 6 goldfish. You go to the pet store and buy 0 new fish. How many goldfish are in your aquarium now? Write the addition equation.

Exercise 3: Explain in your own words why adding zero to a number leaves the number unchanged.

Exercise 4: What is the special mathematical name for the rule that says adding zero to any number equals that number?

Solutions and Explanations

Solution to Exercise 1: 8 + 0 = 8 0 + 5 = 5 0 + 0 = 0 10 + 0 = 10

Solution to Exercise 2: There are still 6 goldfish in your aquarium. The addition equation is 6 + 0 = 6.

Solution to Exercise 3: Adding zero leaves the number unchanged because zero means "none" or "nothing." When you add nothing to an existing collection of objects, the collection does not gain any new items, so the count stays exactly the same.

Solution to Exercise 4: The special mathematical name for this rule is the Identity Property of Zero in Addition.