Lesson 1 of 12 · 8 min

Why new numbers are needed

NCERT §4.1

The club's launcher throws a tennis ball straight up at 20 m/s, so its height after t seconds is h = 20t − 5t² metres. A banner hangs 25 m up. Asha asks when the ball will touch it. The algebra gives an honest answer, but it is not a real number.

The story this chapter follows: Robotics club day in the courtyard

Imagine the school robotics club running its rover and a ball launcher in the courtyard. The courtyard is marked with a 1 m grid and the rover's base is at the origin; east is the real axis and north is the imaginary axis, so the rover standing 3 m east and 4 m north is at 3 + 4i. The launcher throws a tennis ball straight up at 20 m/s, taking g = 10 m/s², so its height is h = 20t − 5t². The numbers are chosen for easy arithmetic, and they carry every example in this chapter.

The lesson in notes

In short

No real number squares to a negative number, because the square of every real number is zero or positive. So x² + 1 = 0, which asks for x² = −1, has no solution among the real numbers.

The same wall appears in every quadratic ax² + bx + c = 0 whose discriminant D = b² − 4ac is negative. For x² − 4x + 5 = 0, D = 16 − 20 = −4 < 0, and the quadratic formula would need the square root of −4.

The graph shows the same thing: y = x² + 1 has its lowest point at (0, 1), so the parabola never touches the x-axis and has no real zero. A quadratic with D < 0 and a > 0 lies wholly above the x-axis.

The way out is to build a larger number system in which some number squares to −1. The real numbers stay inside it unchanged, so every real result still holds.

Historical note: Mahavira (850) stated clearly that a negative quantity is not a square and so has no square root, and Bhaskara said the same in his Bijaganita (1150).

Cardan (1545) met x + y = 10, xy = 40, whose solutions are 5 + √−15 and 5 − √−15, and dismissed them as useless. Later, Albert Girard (about 1625) accepted such roots, the letter i for √−1 is due to Euler, and around 1830 W. R. Hamilton defined a + ib purely as an ordered pair (a, b) of reals.