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The **additive identity** is the number $$0$$. When $$0$$ is added to any number, the value of that number remains unchanged. In formal terms, for any number $$a$$: $$a + 0 = a$$ and $$0 + a = a$$. The name "identity" reflects the idea that the number keeps its identity — it stays the same — after the addition. Zero is the only number with this property, which is why it is uniquely called the additive identity.

Additive Identity

In a corporate accounting ledger, adding a line item with a value of $0 to an account balance does not change the total. What is the formal mathematical term for the number 0 in this context, and based on the concept of addition, why is the word 'identity' used in this name?

The Role of Zero in Financial Adjustments

The additive inverse of a number is its opposite—the value that, when added to it, yields the additive identity, $$0$$. For any real number $$a$$, its additive inverse is $$-a$$, and their defining relationship is $$a + (-a) = 0$$. For instance, the additive inverse of $$5$$ is $$-5$$ since $$5 + (-5) = 0$$, while the additive inverse of $$-3$$ is $$3$$ since $$-3 + 3 = 0$$. Combining a number with its additive inverse reverses the effect of addition, effectively returning the sum to $$0$$.

Additive Inverse

Which equation correctly shows the Additive Identity property for the variable $$A$$?

Since $$5 + 0 = 5$$, the number $$0$$ is called the:

Which number is the additive identity, meaning $$a + 0 = a$$ for any number $$a$$?

Question:
What is the additive identity? Write the property using $$a$$, and explain why it is called an 'identity'.

Sample answer:
The additive identity is $$0$$. For any number $$a$$: $$a + 0 = a$$ and $$0 + a = a$$. It is called an 'identity' because adding $$0$$ does not change the number — it keeps its original value.

Key points:
- The additive identity is 0.
- $$a + 0 = a$$
- $$0 + a = a$$
- 'Identity' means the number's value stays the same after adding 0.
- 0 is the only number with this property.

Rubric:
Full credit: names 0, states the property correctly with $$a$$, and explains why 'identity' fits. Partial credit: gets two of the three parts. No credit: missing or incorrect on all parts.

What is the additive identity? Write the property using $$a$$, and explain why it is called an 'identity'.

Match each term about the additive identity to its meaning.

$$\text{True or False: } 7 + 0 = 7 \text{ because } 0 \text{ is the additive identity.}$$

$$0$$ is called the _____ identity, since $$a + 0 = a$$ for any number $$a$$.

Case context:
$$7 + 0 = 7$$

Question:
What is $$0$$ called, and why does the sum stay the same?

Sample answer:
$$0$$ is the additive identity. Adding $$0$$ does not change a number.

Key points:
- $$0$$ is the additive identity.
- Adding $$0$$ does not change a number.
- For any number $$a$$, $$a + 0 = a$$ and $$0 + a = a$$.

Rubric:
Full credit: Names $$0$$ as the additive identity and states that adding $$0$$ does not change a number. Partial credit: Gives only one of these ideas. No credit: Gives neither idea correctly.

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