Identities

The idea

Two algebraic expressions are equal if they take the same values whenever their letter-numbers are replaced by any integers; a mathematical statement expressing the equality of two algebraic expressions is called an identity.

Is one check enough?

Rani claims a(b + 8) = ab + 8a. She tries a = 2 and b = 3.

Left: 2 × (3 + 8) = 2 × 11 = 22 Right: 2 × 3 + 8 × 2 = 6 + 16 = 22

Both sides match. Can Rani be sure the claim is true for every a and b?

• Yes, they matched
• No, one pair is not enough
• She must test every pair of integers

Test the claims

Pick integers for the letters and press Test. A claim is broken by one pair that does not match. Test every claim.

Identity

Two algebraic expressions are equal if they take the same values whenever their letter-numbers are replaced by any integers. A statement that two expressions are equal like this is called an identity.

a(b + 8) = ab + 8a (a + 1)(b − 1) = ab + b − a − 1 are identities. But a(b + 8) = ab + 8 is true only for a = 1, so it is not an identity.

One pair that breaks a claim shows it is not an identity. Pairs that match only say "equal so far". To be sure, use the distributive property.

Notes

An identity is an equality of two algebraic expressions that holds for every integer value of the letters. One breaking pair shows an equality is not an identity.

Check yourself

Which of these is an identity?

The identity a(b + 8) = ab + 8a holds for negative integers too. Find the value of a(b + 8) when a = −2 and b = 5.

Answer: -26

−2 × (5 + 8) = −2 × 13 = −26. Check with the other side: −2 × 5 + 8 × (−2) = −10 − 16 = −26.

Tom tests a(b + 4) = ab + 4 with a = 1, b = 5. Both sides are 9. Can he say it is an identity?

How can we be sure that an equality is an identity?

• a(b + 3) = ab + 3. Take a = 2 and b = 1: 2 × 4 = 8 but 2 × 1 + 3 = 5. This pair breaks it.
• (a + 1)b = ab + 1. Take a = 2 and b = 3: 3 × 3 = 9 but 2 × 3 + 1 = 7. This pair breaks it.
• a(b + 3) = ab + 3a — correct. Yes! The distributive property gives a(b + 3) = ab + 3a for all integers.
• Yes, both sides matched. One matching pair does not prove it for all integers. Other pairs may break it.
• No. One pair is not enough: for example a = 2, b = 5 gives 18 and 14 — correct. Yes! 2 × 9 = 18 but 2 × 5 + 4 = 14. One breaking pair shows it is not an identity.
• No, because a = 1 is not allowed. a = 1 is a perfectly good integer. The claim just does not hold for all of them.
• Test it with a million pairs of numbers. A million matches still leave other pairs untested. Only algebra can cover all integers.
• Show with the distributive property that both sides become the same expression — correct. Yes! If expanding turns both sides into the same expression, they agree for every integer.
• Test it once with a = 0. a = 0 is just one pair. Many false claims pass at a = 0.