Imaginary Numbers

 

negative result.

An Imaginary Number, when squared , gives aresult.

Try

Let’s try squaring some numbers to see if we can get a negative result:

• 2 × 2 = 4
• (−2) × (−2) = 4 (because a negative times a negative gives a positive)
• 0 × 0 = 0
• 0.1 × 0.1 = 0.01

No luck! Always positive, or zero.

It seems like we cannot multiply a number by itself to get a negative answer …

… but imagine that there is such a number (call it i for imaginary) that could do this:

i × i = −1

Would it be useful, and what could we do with it?

Well, by taking the square root of both sides we get this:

Which means that

i

is the answer to the square root of −1.

Which is actually very useful because …

… by simply accepting that i exists we can solve things
that need the square root of a negative number.

Let us have a go:

Example: What is the square root of −9 ?

√(−9)

= √(9 × −1)

 

= √(9) × √(−1)

 

= 3 × √(−1)

 

= 3i

(see how to simplify square roots)

Hey! that was interesting! The square root of −9 is simply the square root of +9, times i.

In general:

√(−x) = i√x

So long as we keep that little “i” there to remind us that we still
need to multiply by √−1 we are safe to continue with our solution!

Using i

Example: What is (5i)2 ?

(5i)2

= 5i × 5i

 

= 5× 5× i × i

 

= 25 × i2

 

= 25 × −1

 

= −25

Interesting! We used an imaginary number (5i) and ended up with a real solution (−25).

Imaginary numbers can help us solve some equations:

Example: Solve x2 + 1 = 0

Using Real Numbers there is no solution, but now we can solve it!

Subtract 1 from both sides:

x2 = −1

Take the square root of both sides:

x = ± √(−1)

x = ± i

Answer: x = −i or +i

Check:

• (−i)2 + 1 = (−i)(−i) + 1 = +i2 + 1 = −1 + 1 = 0
• (+i)2 +1 = (+i)(+i) +1 = +i2 +1 = −1 + 1 = 0

Unit Imaginary Number

The square root of minus one √(−1) is the “unit” Imaginary Number, the equivalent of 1 for Real Numbers.

In mathematics the symbol for √(−1) is for imaginary.

Can you take the square root of −1?
Well i can!

But in electronics they use

j

(because “i” already means current, and the next letter after i is j).

Examples of Imaginary Numbers

i
12.38i
−i
3i/4
0.01i

π

i

Imaginary Numbers are not

Imaginary Numbers were once thought to be , and so they were called “Imaginary” (to make fun of them).

But then people researched them more and discovered they were actually useful and important because they filled a gap in mathematics … but the “imaginary” name has stuck.

And that is also how the name “Real Numbers” came about (real is not imaginary).

Imaginary Numbers are Useful

 

Complex Numbers

Imaginary numbers become most useful when combined with real numbers to make complex numbers like 3+5i or 6−4i

Spectrum Analyzer

Those cool displays you see when music is playing? Yep, Complex Numbers are used to calculate them! Using something called “Fourier Transforms”.

In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.

It is part of a subject called “Signal Processing”.

 

Electricity

AC (Alternating Current) Electricity changes between positive and negative in a sine wave.

When we combine two AC currents they may not match properly, and it can be very hard to figure out the new current.

But using complex numbers makes it a lot easier to do the calculations.

And the result may have “Imaginary” current, but it can still hurt you!

 

Mandelbrot Set

The beautiful Mandelbrot Set (part of it is pictured here) is based on Complex Numbers.

Quadratic Equation

The Quadratic Equation, which has many uses,
can give results that include imaginary numbers

Also Science, Quantum mechanics and Relativity use complex numbers.

Interesting Property

The Unit Imaginary Number, i, has an interesting property. It “cycles” through 4 different values each time we multiply:

1 × i
 
= i
i × i
 
= −1

−1 × i
 
= −i

−i × i
 
= 1

Back to 1 again!

 

So we have this:

i

= √−1

i2

= −1

i3

= −√−1

i4

= +1

i5

= √−1

i6

= −1
…etc
 

Example What is

i

10 ?

i

10

=

i

4 ×

i

4 ×

i

2

 

=

1

×

1

×

−1

 

=

−1

And that leads us into another topic, the complex plane:

Conclusion

The unit imaginary number, i, equals the square root of minus 1

Imaginary Numbers are not “imaginary”, they really exist and have many uses.

 

 

The Quick Answer

Should you write “than I” or “than me”?

“John is taller than me” and “John is taller than I” are both correct. However, some of your readers might think that “John is taller than me” is wrong (even though it sounds natural), and some of your readers might think that “John is taller than I” sounds pretentious. The safest option is to expand the sentence after the “than.” For example:

• John is taller than I am.

(This construction will satisfy all your readers and remove any ambiguity (more on ambiguity below).)

Should you write “than I” or “than me”?”John is taller than me” and “John is taller than I” are both correct. However, some of your readers might think that “John is taller than me” is wrong (even though it sounds natural), and some of your readers might think that “John is taller than I” sounds pretentious. The safest option is to expand the sentence after the “than.” For example:

Got it? Take a quick test.

Got it? Take a quick test.

“Than I” or “Than Me”?

Authors often ask whether they should write “taller than me” or “taller than I“?

The quick answer is both are correct, but not everyone agrees that both are correct, and that’s the problem. Here’s the issue: the word “than” can be classified as either a

“Than” as a Conjunction

When “than” is used as a

• John is taller than I am.

or

• John is taller than I.
  (This is just a more succinct version.)

“Than” as a Preposition

When “than” is used as a

• John is taller than me.

Grammarians have been arguing for hundreds of years over whether “than” is a conjunction or a preposition. Here’s the bottom line. It is perfectly acceptable to write:

• “than I”

(This means the following are also acceptable: “than he,” “than she,” “than we,” “than they.”)

or

• “than me“

(This means the following are also acceptable: “than him,” “than her,” “than us,” “than them.”)

“Than Me” Sounds More Natural

For most people, the “than me” version sounds more natural than “than I.” However, “than me” is the version that runs the higher risk of being considered wrong. This is almost certainly because the “than I” version has been in use longer and seems more grammatically correct. For some though, the “than I” version sounds pretentious. So, there are some factors to consider before choosing, without any definitive guidelines.

Avoiding Ambiguity with “Than Me”

Unfortunately, it’s not as simple as doing whatever you want and, if challenged, claiming that the world’s grammarians have been squabbling for hundreds of years over this issue. There is another issue to consider. Sometimes, the “than me” version introduces ambiguity. Look at this example:

• John likes Peter more than me.

(This sentence is ambiguous. So, we’ve marked it as wrong.)

This could mean:

• John prefers Peter to me.

or

• John likes Peter more than I like Peter.

The best way to avoid any ambiguity is to treat “than” as a conjunction (i.e., use the “than I” version) and write out the sentence in full. For example:

• John likes Peter more than he likes me.

or

• John likes Peter more than I like Peter.

Than Whom

There is another quirk. Nobody wants to write this:

• You like him more than who?

Everyone agrees it should be:

• You like him more than whom?

So, when “who” is the whom.”

The Final Advice: Expand Your Sentence

The best option is to use the “than I” version and expand the sentence after “than I.” This usually means adding at least the

• John is taller than me.

(This is okay, but some of your readers won’t like it.)

• John is taller than I.

(This is okay, but a few of your readers won’t like it, and some will think it sounds pretentious.)

• John is taller than I am.

(No one can argue this version, and it portrays you as a clear thinker. We’ve given this one a tick.)

Here is an ambiguous example that needs fixing.

• John rates Peter more than I.
• John rates Peter more than me.

(Both of these examples are ambiguous.)

Here are two corrections:

• John rates Peter more than I do.

(No one can argue this version, and it portrays you as a clear thinker.)

• John rates Peter more than he rates me.

(No one can argue this version, and it portrays you as a clear thinker.)

Ready for the Test?

Here is a confirmatory test for this lesson.

This test can also be:

• Edited (i.e., you can delete questions and play with the order of the questions).
• Printed to create a handout.
• Sent electronically to friends or students.

Here is afor this lesson.This test can also be:

Authors often ask whether they should write “taller than” or “taller than”?The quick answer is both are correct, but not everyone agrees that both are correct, and that’s the problem. Here’s the issue: the word “than” can be classified as either a conjunction or a preposition , and that’s the root of the debate.When “than” is used as a conjunction , it looks like this:orWhen “than” is used as a preposition , it looks like this:Grammarians have been arguing for hundreds of years over whether “than” is a conjunction or a preposition. Here’s the bottom line. It is perfectly acceptable to write:orFor most people, the “than” version sounds more natural than “than.” However, “than” is the version that runs the higher risk of being considered wrong. This is almost certainly because the “than” version has been in use longer and seems more grammatically correct. For some though, the “than” version sounds pretentious. So, there are some factors to consider before choosing, without any definitive guidelines.Unfortunately, it’s not as simple as doing whatever you want and, if challenged, claiming that the world’s grammarians have been squabbling for hundreds of years over this issue. There is another issue to consider. Sometimes, the “than” version introduces ambiguity. Look at this example:This could mean:orThe best way to avoid any ambiguity is to treat “than” as a conjunction (i.e., use the “than” version) and write out the sentence in full. For example:orThere is another quirk. Nobody wants to write this:Everyone agrees it should be:So, when “who” is the pronoun in question, “than” should always be treated as a preposition, and you should write “than.”The best option is to use the “than” version and expand the sentence after “than.” This usually means adding at least the verb (e.g., “than I am,” “than I was,” “than they have”). This structure removes all ambiguity and stops your wording sounding pretentious.Here is an ambiguous example that needs fixing.Here are two corrections: