Why the powers of the new symbol run round a cycle of four

What to assume they know

• An equation with no real solution, and the symbol invented to solve it — the relation i² = −1
• Multiplying, then inverting: how division becomes possible — multiplication, the multiplicative inverse, and division defined through it
• Index laws for integer exponents, including negative ones
• Division with remainder, and writing an integer in the form 4k + r

What they should be able to do

• Compute i³, i⁴, i⁵ and i⁶ from the single relation i² = −1
• Explain why the repetition begins at the fourth power and not at the second
• State the four values taken by i raised to an integer power, indexed by the remainder on division by four
• Evaluate i⁻¹ and explain why it equals −i, using the inverse from §4.3.3
• Reduce a large positive exponent to a remainder and read off the value
• Reduce a negative exponent and evaluate it without expanding
• Combine several powers of i in one expression and simplify to the form a + ib

Where it usually goes wrong

• "i to a big power must be a big number." Every integer power of i is one of exactly four values, and none of them is large. Growth is not what repeated multiplication by i does.
• "The pattern repeats every two, because i² = −1." Returning to −1 is not returning to the start. The list repeats only once a power equals 1, and that first happens at the fourth.
• "Even exponents give 1 and odd exponents give i." Parity is the wrong invariant; i² and i⁴ are both even powers and they differ in sign. Only the remainder on division by four decides the value.
• "Negative exponents need a new rule." They need only the inverse of i, which §4.3.3 already supplies. The same four values recur in reverse order.
• "i⁻³⁵ means −i³⁵." The sign belongs to the exponent, not to the base: one is a reciprocal, the other a negation. Take care which exponent you demonstrate this on. At 35 the two happen to agree — both come to i — because an odd power of i is i or −i, and each of those two is its own negative reciprocal. To expose the difference you need an even exponent: with 2 the reciprocal is −1 while the negation is +1, and with 4 the reciprocal is 1 while the negation is −1. The brief's own value for Example 6(ii) is what makes the coincidence at 35 visible.
• "You have to divide the exponent and use the quotient somehow." The quotient is discarded entirely. Everything the exponent contributes is its remainder.

Questions to check understanding

• Evaluate a single large positive or negative power of i
• Simplify a sum or difference of two or more powers of i to the form a + ib
• Find the least positive exponent making a given power equal 1
• Items where a fraction has to be reduced to a power of i before the cycle can be used, as in Miscellaneous Exercise item 14
• One-mark items asking which of four given powers are equal to each other
• Expressions mixing a power of i with an ordinary product, as in Example 2

Examples worth working on the board

Items marked verified are worked out here from the chapter's stated data.

• The printed chain of positive powers (§4.3.5, p. 79): i³ = −i, i⁴ = 1, i⁵ = i, i⁶ = −1. Verified: i³ is i²·i, which is (−1)i; i⁴ is the square of i², which is (−1)², that is 1; i⁵ is i⁴·i and i⁶ is the cube of i².
• The printed chain of negative powers (§4.3.5, p. 79): i⁻¹ = −i, i⁻² = −1, i⁻³ = i, i⁻⁴ = 1. Verified: multiplying 1/i above and below by i gives i/i², which is i/(−1), so i⁻¹ = −i; the rest follow from the same device.
• The general statement (§4.3.5, p. 79, and repeated in the Summary on p. 87). For an integer k, the power with exponent 4k is 1, with 4k+1 is i, with 4k+2 is −1, and with 4k+3 is −i.
• Why the block is four long, not two. Verified: i² is −1, and −1 is not 1, so the list has not yet returned to its starting value; squaring once more is the first point at which it does.
• Example 6(ii) (p. 82). Input: i⁻³⁵. Verified: 35 is 4×8 + 3, so i³⁵ has the same value as i³, which is −i; the reciprocal of −i is i, since (−i)(i) = −i² = 1. So the value is i, which is the value the chapter reaches by a slightly different route on the page.
• Example 2(ii) (p. 80). Inputs exactly as printed: the product of (−i), (2i) and the cube of (−i/8). Verified: the cube of (−i/8) is −i³/512, and since i³ = −i that is i/512; the first two factors give −2i², which is 2; the product is therefore i/256. The chapter prints the same value.
• Exercise 4.1 items driven by the cycle (p. 82). Q2: i⁹ + i¹⁹. Q3: i⁻³⁹. Verified: 9 is 4×2 + 1, so i⁹ = i; 19 is 4×4 + 3, so i¹⁹ = −i; the sum is 0 + i0. For Q3, 39 is 4×9 + 3, so i³⁹ = −i and its reciprocal is i.
• Miscellaneous Exercise Q1 (p. 85). Input: the cube of the sum of i¹⁸ and the twenty-fifth power of 1/i. Verified: 18 is 4×4 + 2, so i¹⁸ = −1; 1/i is −i, and the twenty-fifth power of −i is −(i²⁵), while 25 is 4×6 + 1 so i²⁵ = i, giving −i; the sum inside the bracket is −1 − i. Cubing, (1 + i)² = 2i and (1 + i)³ = 2i + 2i² = −2 + 2i, so the cube of −(1 + i) is 2 − 2i.
• Miscellaneous Exercise Q14 (p. 86). Input: the smallest positive integer m for which the m-th power of (1 + i)/(1 − i) equals 1. Verified: multiplying above and below by 1 + i turns the fraction into 2i/2, which is i, so the question is the smallest positive m with i^m = 1 — and by the cycle that is 4. This is the cleanest single item in the chapter for showing the period doing real work.

All values marked verified are worked out here on the printed items; the chapter prints no answers on these pages.

Figures to have open

• A four-station loop diagram with i, −1, −i and 1 arranged in order, and arrows showing one step per multiplication in one direction and per division in the other. Standard schematic, and the central image of this topic; the chapter has no figure in §4.3.5.
• An exponent strip running from about −6 to +10 with the four values repeating underneath, so the periodicity is visible at a glance. Standard schematic.
• A remainder gadget: any integer entering, 4k + r leaving, only r reaching the loop. Standard schematic.
• No textbook figure is required for this topic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 4, whose printed title is Complex Numbers and Quadratic Equations; §4.3.5, p. 79.
• Example 2 on p. 80 and Example 6, part two, on p. 82.
• Exercise 4.1, items 2 and 3, p. 82.
• Miscellaneous Exercise on Chapter 4, item 1 on p. 85 and item 14 on p. 86.
• Summary, p. 87, which restates the four values for an integer k.