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Real numbers · 2 marks

Consider this puzzle: What is the square root of −1? We know that 1 × 1 = 1. We also know that (−1) × (−1) = 1. There is no Real Number that, when multiplied by itself, results in a negative number. Thus, √−1 cannot exist on number line.

Answer: No real number squares to −1: a positive times a positive is positive, a negative times a negative is positive, and 0 × 0 = 0. So √−1 is not on the real number line. Mathematicians created a new number i with i × i = −1, the start of the imaginary numbers.

Step-by-step solution

Idea: Check every kind of real number: positive, negative and zero. None has a negative square. So answering the puzzle needs numbers off the real line.

  1. If x > 0, then x × x > 0. If x < 0, then x × x > 0 (debt × debt is a fortune). If x = 0, then x × x = 0. So the square of every real number is ≥ 0, never −1.1 mark
  2. Hence no point of the real number line is √−1. To solve the puzzle, mathematicians stepped off the line and defined a new number i with i2 = −1. Numbers built from it are called imaginary numbers; they are used in electrical engineering and physics. So the journey does not end with the real numbers.1 mark
There is no real square root of −1, because the square of every real number is 0 or positive. √−1 is defined as a new, imaginary number i with i² = −1.

Answer to write in the exam

x > 0 ⇒ x2 > 0; x < 0 ⇒ x2 > 0; x = 0 ⇒ x2 = 0

∴ x2 ≥ 0 for every real x; no real x has x2 = −1

∴ √−1 is not a real number; it is the imaginary number i, with i2 = −1.

Common mistakes that cost marks

  • Answering √−1 = −1. But (−1) × (−1) = +1, not −1.
  • Thinking √−1 lies somewhere between −1 and 0 on the number line. Every number there has a positive square.
  • Confusing −√1 = −1 (a real number) with √−1 (not real).

How this can come in the exam

MCQ (1 mark)

Which of the following is NOT a real number?

  1. −√9
  2. √0
  3. √−9
  4. ∛−8
Show answer

(C) √−9
−√9 = −3, √0 = 0 and ∛−8 = −2 (since (−2)3 = −8) are real; no real number squares to −9.

Assertion–Reason (1 mark)

Assertion (A): There is no real number x with x2 = −4.
Reason (R): The square of any real number is non-negative.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
R is true for positive, negative and zero values, and it is exactly why A holds.

Try one yourself

For which real numbers x is √(x − 3) a real number?

Show answer

When x − 3 ≥ 0, i.e. x ≥ 3.

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