The real number line stretches infinitely in both directions, accommodating every integer, fraction, and irrational value we encounter in daily life. Yet certain equations expose its limitations. The simple expression x2=−1 has no home among real numbers — squaring any real value, positive or negative, always produces a non-negative result. To address this gap, mathematics extends beyond the familiar line into a richer structure built upon a single new element: the imaginary unit i.
Every number system so far was built to answer a question the one before could not.
Every expansion of our number system arose from an equation that refused to yield solutions. Negative numbers emerged when subtraction demanded answers beyond the counting numbers. Fractions appeared when division insisted on exactness. Irrational numbers surfaced when geometry required lengths like 2 that no fraction could capture. Each time, mathematicians faced a choice: declare certain problems unsolvable, or expand the boundaries of what counts as a number.
Five systems, each forced by a problem the one before could not answer: the counting numbers N cannot give 3−5, the integers Z cannot give 1÷3, the rationals Q cannot give the diagonal of a unit square, 2, and the reals R cannot solve x2+1=0. The complex numbers C can. Read the chain downward; each arrow carries the problem that forced the next step.
The equation x2+1=0 poses exactly this challenge to the real number system R. Rearranging gives x2=−1, which asks: what number, when multiplied by itself, produces negative one? No real number qualifies. Positive numbers squared remain positive. Negative numbers squared also yield positive results, since two negative signs cancel. Zero squared gives zero. The entire real line offers no candidate.
x
x²
equals −1?
−2
4
no
−1
1
no
−0.5
0.25
no
0
0
no
0.5
0.25
no
1
1
no
2
4
no
Every square in the table is 0 or positive, and no real number could do better: a positive times a positive is positive, a negative times a negative is positive too.
The parabola y=x2 never dips below the axis, so it never meets the line y=−1: the equation has no real solution.
The definition
The imaginary uniti is a number defined by the single property i2=−1. An imaginary number is a real multiple of it, bi with b real and b=0.
So x2=−1 has the two solutions x=i and x=−i, and −9=3i.
An imaginary number is a number, but not one on the real number line.
Rather than abandon such equations as meaningless, mathematicians introduced a new object: the imaginary unit, denoted i, defined by the single property i2=−1. This definition creates exactly what the equation demands — a number whose square is negative one. The symbol i does not represent any point on the traditional number line; it exists perpendicular to all real numbers, opening an entirely new dimension for mathematical exploration.
An equivalent notation writes i=−1, interpreting i as the principal square root of negative one — both ways of writing the unit appear among the complex number symbols. This form emphasizes the connection to familiar radical expressions, though it requires caution. The usual rules for manipulating square roots do not always extend safely to negative radicands, a subtlety explored in the next section. Despite its name, nothing about the imaginary unit is fictitious or unreal — it is simply a different kind of number, as legitimate as any other, governed by consistent rules that extend arithmetic into new territory.
The old rules of arithmetic still hold.
Treat i like any other symbol and replace i2 by −1 wherever it appears: everything else works as before.
old rule
with i
like terms add
2i + 3i = 5i
numbers multiply
3 · 2i = 6i
a square root undoes a square
√−9 = 3i, and (3i)² = 9i² = −9
x² = −1 now has roots
x = i and x = −i
Nothing is lost on the way down. Each system keeps the arithmetic of the one above it and adds the missing answers, so the complex numbers extend the reals rather than replace them: a real number is a complex number whose imaginary part is zero. What the chain also shows is that i is not a stranger act than the others; admitting 2 was once the same kind of step.
What you lose: order, and one rule for square roots.
Admitting i costs two familiar habits.
familiar rule
what happens with i
every number is > 0, = 0 or < 0
i is none of them: i > 0 and i < 0 would both give i² > 0
√a · √b = √(ab)
√−1 · √−1 = i · i = −1, but √((−1)(−1)) = √1 = 1
So imaginary numbers cannot be compared with <, and square roots of negative numbers are best rewritten with i before multiplying: −4⋅−9=2i⋅3i=−6, not 36=6.
The rest of the page covers square roots of negative numbers, pure imaginary numbers, notation, the imaginary part of a complex number, the powers of i and their cycle, arithmetic with imaginary numbers, and their place in the complex plane.
Simplifying Square Roots of Negative Numbers
With the imaginary unit established, we gain a systematic method for handling square roots of negative numbers — expressions that previously had no meaning within R. The technique extracts the negative sign as a factor of i, leaving behind an ordinary real radical.
The rule states: for any positive real number a, the square root of its negative equals i times the square root of the positive value:
Consider −4. Applying the rule: −4=i4=i⋅2=2i. The answer 2i satisfies verification, since (2i)2=4i2=4(−1)=−4.
For −7, where the radicand has no perfect square factor, we obtain −7=i7, leaving the irrational portion intact. A more involved example: −12=i12=i4⋅3=2i3, combining the extraction of i with standard simplification of the remaining radical.
Fact: Even Euler multiplied the roots
In his 1770 algebra textbook, Euler wrote −1⋅−4=4=2. Taking out i first gives i⋅2i=2i2=−2: the product rule for roots misled even him.
One critical warning accompanies this technique: the familiar product rule for radicals, a⋅b=ab, fails when both radicands are negative. A naive calculation might proceed as −1⋅−1=(−1)(−1)=1=1. But this contradicts the definition — we know −1=i, and i⋅i=i2=−1, not +1. The error arises because the product rule assumes non-negative radicands. When negatives appear, extract i from each factor first, then multiply: −1⋅−1=i⋅i=i2=−1. This trap catches many students; the safest practice converts all negative square roots to i-form before performing any multiplication.
Expression
Method
Result
√(−4)
i · √4
2i
√(−7)
i · √7 (radicand has no perfect square factor)
i√7
√(−12)
i · √12 = i · 2√3
2i√3
√(−1) · √(−1) (correct)
extract i first: i · i = i²
−1 ✓
√(−1) · √(−1) (wrong)
apply product rule: √((−1)(−1)) = √1
1 ✗— product rule fails for negative radicands
Defining Pure Imaginary Numbers
The imaginary unit i serves as the foundation for an entire family of numbers. Multiply i by any real number b, and the result bi belongs to a special category: the pure imaginary numbers. These occupy a distinct position within the broader complex number system, characterized by the complete absence of a real component.
Formally, when we write a complex number in standard form z=a+bi, the number qualifies as pure imaginary precisely when a=0. The real part vanishes entirely, leaving only the imaginary term. The structure reduces to z=0+bi=bi, where b can be any real value — positive, negative, rational, irrational, or even zero.
Pitfall: Judging by the letter i
Calling 4i2 pure imaginary because it contains i skips the simplification: 4i2=4(−1)=−4, a real number. Write the number as a+bi first; it is pure imaginary only when a=0.
Examples span a wide range of appearances. Simple cases include 3i, −7i, and 21i. More elaborate forms arise in calculations: 5i carries an irrational coefficient, while 1363i combines fractions and radicals. Despite their varied appearances, all share the defining characteristic — no real part exists alongside the imaginary term.
The number zero occupies a unique position in this classification. Written as 0=0+0i, it satisfies the condition for being real (the imaginary part equals zero) and simultaneously satisfies the condition for being pure imaginary (the real part equals zero). No other number achieves this dual membership. Zero stands alone at the intersection of the real and imaginary worlds, belonging fully to both categories. On the complex plane, this corresponds to the single point where the horizontal real axis and the vertical imaginary axis cross — the origin itself.
z = 2i: a point on the imaginary axis
The number has no real part, so it sits directly on the vertical axis at height 2; every pure imaginary number lives on this axis and nowhere else, just as every real number lives on the horizontal one. Slide the point along the axis and watch the real part stay at 0 on the complex number explorer.
The imaginary axis is the number line of the pure imaginary numbers.
Imaginary Unit Notation
Notation
Imaginary Unit Notation
Basics named the number; this page owns the unit itself — the letter, its radical alias, and the coefficient conventions around it. Engineering's dissent and the index-i collision live here too.
One letter defines the whole system: i is the number whose square is −1. Typography carries meaning here — ISO and physics style guides set the constant upright (i) to separate it from italic variables, though most textbooks, this site included, keep the italic.
CasesCoefficient 1 is never written — i, not 1i — and parenthesization decides what a square hits: (2i)2=−4, while 2i2=−2.
Also writtenj — electrical engineering's replacement, with i reserved for current: circuit texts write a+jb, and Python's complex literals are 3+2j.
Do not confuseThe index i of sums and matrix entries — ai, ∑i, aij — a pure counter with no square rule; texts mixing both switch the counter to k, and the sequence subscript page owns that job.
Same glyph elsewhereThe unit vector i of cross products — bold, and no accident: Hamilton's quaternion units i,j,k fathered both notations.
−1
the square root of minus one
The radical spelling of the unit: i=−1, read as the principal root. A definition by picture rather than by property — rigorous treatments start from i2=−1 instead, because the radical sign imports rules that do not all survive the trip.
CasesSafe as a name, dangerous as an operand: −a=ia converts once, then all arithmetic happens in i-form — the working rule of Simplifying Square Roots of Negative Numbers above.
Also written(−1)1/2 — the fractional-exponent spelling, native wherever rational exponents do the writing.
Do not confuseA license to multiply radicals. −1⋅−1=1=1 looks legal and is false — the product rule ab=ab demands non-negative radicands, and the true value is i⋅i=−1.
bi
b i — a pure imaginary number
The pure-imaginary spelling of the general form: real part gone, coefficient before unit — 2i, never i2. Defining Pure Imaginary Numbers below fixes when the name applies.
CasesRadical coefficients flip the order: i7 is preferred to 7i, keeping the unit visibly outside the radicand; the same i-first instinct writes cosθ+isinθ.
Do not confuseThe imaginary part. Im(2−7i) is −7, the bare coefficient — not −7i; Critical Distinction below exists for exactly this error.
Critical Distinction: The "Imaginary Part" is Real
The terminology surrounding complex numbers contains a paradox that trips up countless students. In the expression z=a+bi, we call b the "imaginary part" — yet b itself is an ordinary real number. The name refers to which component of z the value describes, not to the nature of b as a number.
Consider z=5+3i. The real part equals 5, and the imaginary part equals 3. Not 3i — just 3. The coefficient standing before i is what we extract and label as the imaginary part. When asked for Im(z), the correct response is the real number 3, stripped of its accompanying i. The imaginary unit indicates where the value lives within the complex structure; it does not become part of the value itself.
Notation formalizes this distinction. For any complex number z=a+bi, we write Re(z)=a for the real part and Im(z)=b for the imaginary part.
RULE
Both Re(z) and Im(z) produce real number outputs — they extract coordinates from the complex number, much like obtaining the x and y values from a point in the plane.
The confusion intensifies with negative imaginary parts. Given z=2−7i, students often struggle with whether the imaginary part is −7 or 7. Writing the number in strict standard form as z=2+(−7)i clarifies the situation: the coefficient multiplying i is −7, so Im(z)=−7. The sign belongs to the imaginary part, not to the i.
A reliable rule of thumb: when identifying the imaginary part of any complex number, mentally rewrite it in the form a+bi and read off whatever real number occupies the position directly before i. That number — complete with its sign, whether integer, fraction, or irrational — constitutes the imaginary part. The i merely marks the location; it never participates in the answer to "what is Im(z)?"
Im(z) is a signed real number
z = 3 − 2i, written in strict standard form as 3 + (−2)i. The figure reads its two parts as coordinates: a = 3 across, b = −2 down. The imaginary part is that −2 — a real number, sign included, with no i attached. Try other points on the complex number explorer.
Keep the sign and drop the i: Im(2 − 7i) = −7, not 7 and not −7i.
The Cyclic Nature of the Powers of i
Raising i to successive powers produces a pattern that loops back on itself every four steps. This cyclic behavior distinguishes imaginary exponentiation from real number powers and provides a powerful shortcut for calculations that would otherwise seem impossibly complex.
Begin with the first four powers. By definition, i1=i. Squaring gives i2=−1, the foundational property. For the cube, multiply once more: i3=i2⋅i=(−1)⋅i=−i. The fourth power completes the cycle: i4=i2⋅i2=(−1)(−1)=1. And now the pattern resets — i5=i4⋅i=1⋅i=i, returning us to where we started.
The sequence i,−1,−i,1 repeats indefinitely. Every fourth power brings the value back to 1, launching the cycle anew. This periodicity means that i17 equals i1, that i42 equals i2, and that i1000 equals i4=1. No matter how large the exponent grows, only four possible outcomes exist.
Pitfall: Treating every even power as −1
Taking i100=−1 because 100 is even treats the powers like (−1)n, which alternate in twos. The powers of i cycle in fours: i100=(i4)25=1; only even exponents with remainder 2, like i102, give −1.
The computational shortcut exploits this four-cycle through division with remainder. To evaluate ik for any positive integer k, divide k by 4 and examine the remainder r. The answer matches ir: remainder 0 yields 1, remainder 1 yields i, remainder 2 yields −1, and remainder 3 yields −i.
Example
Consider i323. Dividing 323 by 4 produces 80 with remainder 3. Therefore i323=i3=−i.
The enormous exponent collapses to a single-digit calculation. Similarly, i500 has remainder 0 when divided by 4, so i500=i0=1. For i67, the division 67÷4 leaves remainder 3, giving i67=−i.
Pitfall: Ignoring the sign of the exponent
Evaluating i−3 as i3=−i drops the minus sign of the exponent. In fact i−3=i31=−i1=i; adding 4 to the exponent gives the same, i−3=i1=i.
Negative exponents follow the same logic after one preliminary step. Recall that i−1=i1, which simplifies by multiplying numerator and denominator by i: i1⋅ii=i2i=−1i=−i. Thus i−1=−i, corresponding to remainder 3 in the cycle. The pattern for negative powers runs backward through the same four values, and the remainder method adapts naturally once the exponent's sign is handled.
n mod 4
iⁿ
Examples
0
1
i⁰ = 1, i⁴ = 1, i⁸ = 1, i⁵⁰⁰ = 1
1
i
i¹ = i, i⁵ = i, i⁴⁰¹ = i
2
−1
i² = −1, i⁶ = −1, i¹⁰² = −1
3
−i
i³ = −i, i⁷ = −i, i³²³ = −i, i⁻¹ = −i
The powers of i: a cycle of four
The four values i, −1, −i, 1 sit on a loop and each multiplication by i steps one place round it, so i⁵ lands back on i: the remainder of the exponent on division by 4 is all that matters. The highlighted node is the current power. Step the exponent up and watch the cycle repeat on the powers of i visualizer.
Geometrically each multiplication by i is a quarter turn, which is why the cycle closes after four.
Geometric Visualization: The Imaginary Axis
Fact: Gauss would have called it lateral
Descartes named such roots imaginary in 1637, and the name stuck. In 1831 Gauss wrote that calling +1, −1 and −1 the direct, inverse and lateral units would have removed the mystery: i simply points sideways.
Numbers become tangible when we can see them. The real numbers arrange themselves along a horizontal line — the familiar number line from elementary mathematics, stretching from negative infinity on the left through zero to positive infinity on the right. Pure imaginary numbers demand their own territory, and they claim the perpendicular direction: a vertical axis passing through zero.
The complex plane combines these two axes into a unified coordinate system. The horizontal axis carries real numbers, while the vertical axis carries pure imaginary numbers. Every point in this plane represents a complex number, with its horizontal position indicating the real part and its vertical position indicating the imaginary part.
Pure imaginary numbers occupy the vertical axis exclusively. The number 3i sits three units above the origin. The number −2i lies two units below. The value 21i hovers halfway between zero and i. Each pure imaginary number bi corresponds to the ordered pair (0,b) — zero horizontal displacement, vertical displacement equal to b. The first coordinate vanishes because pure imaginary numbers have no real part.
This geometric arrangement reveals a beautiful symmetry in the structure of complex numbers. Real numbers provide the "width" of the system, extending left and right from the origin. Imaginary numbers provide the "height," extending up and down. Together they span every direction, allowing complex numbers to fill the entire plane rather than merely a line. A general complex number a+bi appears at coordinates (a,b), combining horizontal and vertical components.
The origin deserves special attention in this picture. Located at (0,0), it represents the number 0=0+0i. This single point lies on both axes simultaneously — it belongs to the real numbers (being located on the horizontal axis) and to the pure imaginary numbers (being located on the vertical axis). No other point shares this property; every other location commits to one category or exists off both axes entirely as a genuinely complex number with nonzero parts.
Pure imaginary numbers form a second number line at right angles to the real one, meeting it only at 0.
Mathematical Properties and Conjugates
Pure imaginary numbers exhibit distinctive behavior under the standard operations of complex arithmetic, particularly when interacting with the conjugate operation. Understanding these properties illuminates why pure imaginaries form a coherent subcategory within C.
For a general complex number z=a+bi, the conjugate zˉ=a−bi negates only the imaginary component while preserving the real part. When applied to a pure imaginary number z=bi (where a=0), this operation produces zˉ=0−bi=−bi. The conjugate of a pure imaginary is simply its negative. Geometrically, reflection across the real axis maps a point on the positive imaginary axis to the corresponding point on the negative imaginary axis — a straightforward flip through the origin along the vertical line.
This observation leads to a powerful classification theorem: a complex number is pure imaginary if and only if zˉ=−z. The condition zˉ=−z expands to a−bi=−a−bi, which forces a=−a and therefore a=0. Conversely, when a=0, the equality zˉ=−bi=−z holds automatically. This test provides an algebraic criterion for identifying pure imaginary numbers without examining their form directly.
Arithmetic with conjugates produces predictable outcomes for any complex number. Adding z to its conjugate eliminates the imaginary part: z+zˉ=(a+bi)+(a−bi)=2a, always a real number. Subtracting the conjugate from the original eliminates the real part: z−zˉ=(a+bi)−(a−bi)=2bi, always pure imaginary. For pure imaginary inputs where a=0, these formulas specialize: z+zˉ=0 and z−zˉ=2bi=2z.
The absolute value of a pure imaginary number simplifies dramatically. For general z=a+bi, the modulus formula gives ∣z∣=a2+b2. With a=0, this reduces to ∣bi∣=0+b2=b2=∣b∣. The modulus of a pure imaginary number equals the absolute value of its coefficient — the distance from the origin along the vertical axis. The number 5i has modulus 5; the number −3i has modulus 3. No square root computation required, just the magnitude of the real coefficient stripped of its sign.
Operation on z
General z = a + bi
Pure imaginary z = bi (a = 0)
Conjugate z̄
a − bi
−bi = −z
z + z̄
2a (real)
0
z − z̄
2bi (pure imaginary)
2bi = 2z
Modulus |z|
√(a² + b²)
|b|
Re(z), Im(z)
a, b
0, b
Location in complex plane
point (a, b) anywhere in the plane
point (0, b) on the vertical (imaginary) axis
Summary: Four Equivalent Ways to Recognize a Pure Imaginary Number
Pure imaginary numbers can be recognized in several equivalent ways — algebraically through their standard form, by a conjugate test, geometrically by their location on the imaginary axis, or multiplicatively as products of i with a real number. The table below collects these four characterizations side by side, applying each to a representative pure imaginary number and a non-example to make the criteria concrete.
Characterization
Form / test
Pure imaginary example: 5i
Non-example: 7
Algebraic (standard form)
Re(z) = 0; written as bi with b ∈ ℝ
Re(5i) = 0 ✓
Re(7) = 7 ≠ 0 ✗
Conjugate test
z̄ = −z
conjugate of 5i is −5i = −(5i) ✓
conjugate of 7 is 7 ≠ −7 ✗
Geometric
lies on vertical (imaginary) axis
point (0, 5) on imaginary axis ✓
point (7, 0) on real axis ✗
Multiplicative
z = i · b for some real b
5i = i · 5 with b = 5 ✓
7 = i · b has no real solution ✗
Common Pitfalls
Six mistakes come up again and again in first work with imaginary numbers. Four are marked where they happen in the sections above; the other two, multiplying negative radicands under one root and keeping the i in the imaginary part, are the ones the page already warns about in its own text. Here each is worked through in full, with the wrong and the right working side by side.
Taking the minus sign out of the root
The square root of a negative number is simplified by moving the minus sign in front of the root, as if −a were −a.
Wrong working: −9=−9=−3.
Right working: the minus sign becomes the factor i, so −9=i9=3i.
Why it happens: the −1 inside the root looks like an ordinary factor that can be pulled out, the way a factor 4 comes out as 2. But −1 is i, not −1.
How to check: square the answer. (−3)2=9, which is not −9, while (3i)2=9i2=−9.
Multiplying negative radicands under one root
Two square roots of negative numbers are combined into a single root with the product rule a⋅b=ab, which holds only for non-negative radicands.
Wrong working: −4⋅−9=(−4)(−9)=36=6.
Right working: convert each root to i-form first, then multiply: −4⋅−9=2i⋅3i=6i2=−6.
Why it happens: the product rule is used so often with positive numbers that its condition is forgotten.
How to check: the product of two pure imaginary numbers bi and ci is bc⋅i2=−bc. Since −4=2i and −9=3i both have positive coefficients, their product must be negative, so 6 cannot be right.
Judging by the letter i
A number is called pure imaginary because i appears in it, without first writing it in the form a+bi.
Wrong working: 4i2 contains i, so it is pure imaginary.
Right working: 4i2=4(−1)=−4=−4+0i. Its real part is −4, not 0, so it is a real number, not a pure imaginary one. The reverse also happens: (1+i)2=1+2i+i2=2i is pure imaginary, although it was built from a number with real part 1.
Why it happens: the letter i is the visible mark of imaginary numbers, and i2 still shows it even though its value is −1.
How to check: simplify fully, using i2=−1, until the number reads a+bi with real a and b. Then look only at a.
Keeping the i in the imaginary part
Asked for Im(z), the whole imaginary term is given, i included, or its sign is dropped.
Wrong working: for z=4−6i, Im(z)=−6i, or Im(z)=6.
Right working: write z=4+(−6)i. The real number standing before i is −6, so Im(z)=−6.
Why it happens: the name "imaginary part" suggests an imaginary number, and in 4−6i the minus sign looks like part of the subtraction rather than part of the coefficient.
How to check: Im(z) is always a real number, so an answer containing i is wrong. Rebuilding z as Re(z)+Im(z)⋅i=4+(−6)i must give back the original number.
Treating every even power as −1
Every even power of i is taken to be −1 and every odd power to be i, as if the powers alternated in twos like the powers of −1.
Wrong working: 100 is even, so i100=−1.
Right working: the powers of i repeat every four steps. Since 100=4⋅25, the remainder is 0 and i100=(i4)25=125=1. An even exponent gives −1 only when its remainder on division by 4 is 2, as in i102=−1.
Why it happens: the first fact learned is i2=−1, and it is extended to all even exponents without checking i4=1.
How to check: divide the exponent by 4 and use the remainder, never just the parity. For 100 the remainder is 0, so the answer is 1.
Ignoring the sign of the exponent
For a negative exponent, the minus sign is dropped and the remainder method is applied to the positive number.
Wrong working: i−3=i3=−i.
Right working: i−3=i31=−i1=−i1=−(−i)=i. Adding 4 to the exponent does not change the value, so i−3=i−3+4=i1=i as well.
Why it happens: the remainder method is learned for positive exponents, and −3 is read as 3.
How to check: multiply the answer by i3. A correct i−3 gives i−3⋅i3=i0=1: here i⋅(−i)=−i2=1. The wrong answer gives (−i)(−i)=i2=−1.
Imaginary Numbers FAQ
What is the imaginary unit i?
+
It is the number defined by the property that its square equals −1. No real number does this, since squaring any real gives a non-negative result, so i is introduced as a new object rather than discovered among existing numbers. Everything else about imaginary and complex numbers is built from this single definition.Read more →
Why doesn't the rule √a · √b = √(ab) work for negative numbers?
+
Because that rule requires both values to be non-negative. Ignoring the condition gives √(−1)·√(−1) = √1 = 1, when the correct answer is −1. Convert each negative root to the form bi first, then multiply. Skipping that conversion is the most common error with square roots of negatives.Read more →
How do you calculate i to a large power?
+
Divide the exponent by 4 and keep only the remainder, since the powers repeat in a cycle of four: i, −1, −i, 1. A remainder of 0 gives 1, of 1 gives i, of 2 gives −1, and of 3 gives −i. So i to the 2026 has remainder 2 and equals −1. Watch the cycle turn on the powers of i explorer.Read more →
Is zero a pure imaginary number?
+
It depends on the convention, which is why the question causes trouble. Zero can be written as 0i, so it satisfies the form, but many texts require a pure imaginary number to have a non-zero imaginary part and exclude it. Zero is the only number that is simultaneously real and imaginary under the looser definition.Read more →
What are imaginary numbers actually used for?
+
They describe anything that rotates or oscillates. Alternating current, signal processing, control systems, and quantum mechanics all run on them, because rotation in the plane is exactly what multiplication by a complex number does. The name is a historical accident from a period when mathematicians distrusted them; nothing about them is less real than negative numbers.Read more →
Why are imaginary numbers called imaginary?
+
Because Descartes called such roots imaginary in 1637, and the name stuck long after the numbers were accepted. Gauss thought it a poor choice: in 1831 he wrote that calling +1, −1 and −1 the direct, inverse and lateral units would have removed the mystery. His name describes the picture, since i simply points sideways, at right angles to the real number line.Read more →
How do you work out i to a negative power?
+
Add 4 to the exponent as many times as needed, since that never changes the value: i−1=i3=−i and i−3=i1=i. Directly, i−1=i1=i2i=−i. The usual error is dropping the minus sign and taking i−3 to be i3=−i. To check, multiply: i−3⋅i3 must be 1, and indeed i⋅(−i)=−i2=1.Read more →
Why do engineers write j instead of i?
+
Because in electrical engineering i already stands for current, so the imaginary unit is renamed j and circuit texts write a+jb. Python uses the same letter: its complex numbers are typed like 3+2j. Only the letter changes: j2=−1, and every rule on this page holds as written with j in place of i.Read more →
