Number systems common mistakes

Common mistakes

• Leaving zero out of ℕ

  Treating the natural numbers as starting at 1.

  Fix itUse the definition in this specification: ℕ = {0, 1, 2, 3, …}, so zero is included.

• Forgetting that zero is an integer

  Treating 0 as neither positive nor negative and therefore assuming it is not in ℤ.

  Fix itRemember that 0 is explicitly included in the definition of ℤ, even though it is neither positive nor negative.

• Assuming integers are not rational

  Thinking that a number must be displayed as a fraction to be rational, so an integer such as 7 is excluded.

  Fix itRemember that 7 can be written as 7/1. Since it can be represented as a ratio of integers, it belongs to ℚ.

• Confusing notation with classification

  Assuming that √2 is irrational merely because it uses a square-root symbol, or assuming that a number is not irrational because it is written using a different notation.

  Fix itUse the defining test: an irrational number cannot be written as a fraction. √2 is an example.

• Treating rational and real as separate categories

  Assuming that a number must be either rational or real.

  Fix itRational numbers are a subset of the real numbers. A rational number is therefore also real; irrational numbers are real but not rational.

• Swapping adjacent positions

  Assigning the 3rd position to the fourth object or the 4th position to the third object.

  Fix itRead the set from the beginning and count each object carefully, assigning positions in sequence: 1st, 2nd, 3rd, 4th.

• Using a natural number for every value

  Choosing a natural number for a measurement simply because the measurement is currently a whole number.

  Fix itIdentify the purpose of the value. Counts use natural numbers, while measurements use real numbers because measurements may need fractional values.