The Square Root Nobody Had

The equation x^2 + 1 = 0 rearranges to x^2 = -1, and no real number squares to a negative result: squaring a positive number gives a positive result, squaring a negative number ALSO gives a positive result, and squaring zero gives zero. So within the real numbers, this equation has no solution at all. Rather than give up, mathematics extends the number system by introducing a new symbol, i, defined by the one property i^2 = -1. A complex number is then any expression of the form a + ib, where a and b are ordinary real numbers. For the complex number z = 7 - i4, the real part, written Re(z), is 7, and the imaginary part, written Im(z), is -4 (just the number multiplying i, not -4i itself).

Two complex numbers a+ib and c+id are equal exactly when their real parts match AND their imaginary parts match: a=c and b=d, both conditions at once. This turns a single complex equation into two ordinary real equations. Suppose 5x + i(2x - 3y) = 10 + i(7), where x and y are real numbers. Equating real parts gives 5x = 10, so x = 2. Equating imaginary parts gives 2x - 3y = 7; substituting x = 2 gives 4 - 3y = 7, so -3y = 3, meaning y = -1. One complex equation, hiding two ordinary ones.

Adding complex numbers is straightforward: add the real parts, add the imaginary parts, separately. (3+i7) + (-5+i2) = (3-5) + i(7+2) = -2 + i9. Multiplying is less obvious, since it uses the distributive property together with i^2=-1: (a+ib)(c+id) expands to ac + iad + ibc + i^2bd, and since i^2=-1, this simplifies to (ac-bd) + i(ad+bc). Checking (4+i3)(2-i5): the real part is (4)(2) - (3)(-5) = 8 + 15 = 23, and the imaginary part is (4)(-5) + (3)(2) = -20 + 6 = -14, giving (4+i3)(2-i5) = 23 - i14. Every familiar arithmetic law, closure, commutativity, associativity, distributivity, carries over unchanged from real numbers to complex numbers.

Since i^2 = -1, it follows that i^3 = i^2 times i = -i, and i^4 = i^2 times i^2 = (-1)(-1) = 1. But then i^5 = i^4 times i = i again, and the whole pattern repeats: i, -1, -i, 1, forever, every four powers. This means any power of i can be simplified just by finding the remainder when the exponent is divided by 4: i^(4k)=1, i^(4k+1)=i, i^(4k+2)=-1, i^(4k+3)=-i, for any whole number k. To simplify i^23, divide 23 by 4: 23 = 4(5) + 3, so i^23 matches the i^(4k+3) case, giving i^23 = -i.

For any positive real number a, the square root of -a is defined as (square root of a) times i, so the square root of -49 is 7i. This lets every negative number get a square root at last. But a familiar real-number rule breaks down here: for positive numbers, square-root-of-a times square-root-of-b always equals square-root-of-ab, but this fails once BOTH numbers are negative. Checking directly: the square root of -9 is 3i, and the square root of -16 is 4i, so their product is (3i)(4i) = 12i^2 = -12. But the square root of the product, square-root-of[(-9)(-16)] = square-root-of-144 = 12. These do not match: -12 is not 12. The safe rule is to convert each negative square root to i-form FIRST, then multiply, rather than combining the numbers under one root first.

The modulus of z = a+ib, written |z|, is the non-negative real number square-root-of(a^2+b^2). For z = 8-i6, |z| = square-root-of(64+36) = square-root-of-100 = 10. The conjugate of z, written z-bar, simply flips the sign of the imaginary part: the conjugate of 8-i6 is 8+i6. Multiplying a complex number by its own conjugate always produces a real number: (8-i6)(8+i6) = 64 - 36i^2 = 64+36 = 100, exactly |z| squared. This trick is exactly how division works: to simplify (7+i)/(1-i2), multiply top and bottom by the conjugate of the denominator, 1+i2: (7+i)(1+i2) divided by (1-i2)(1+i2) = (7+14i+i+2i^2)/(1+4) = (7+15i-2)/5 = (5+15i)/5 = 1+i3.

Every complex number x+iy corresponds to a unique point (x,y) in a plane, called the Argand plane: the horizontal axis is the real axis, and the vertical axis is the imaginary axis. The modulus of a complex number is exactly its distance from the origin in this plane, and the conjugate of a complex number is the mirror image of its point, reflected across the real axis. This idea of treating a number as an ordered pair was made fully rigorous by the Irish mathematician William Rowan Hamilton around 1830, avoiding any mystical talk of 'imaginary' numbers altogether. The struggle to accept these numbers stretches back centuries: the Indian mathematicians Mahavira (around 850 CE) and Bhaskara (in his 1150 CE work Bijaganita) both stated plainly that a negative quantity has no square root, and the Italian mathematician Cardano, encountering expressions like 5 plus the square root of -15 while solving a problem in 1545, dismissed them as 'useless' even while writing them down. It took until Euler introduced the symbol i, and Hamilton gave the idea a rigorous footing, for these numbers to be fully accepted.