Imaginary numbers

Imaginary numbers So let’s all use our imaginations!

Aren’t all numbers “imaginary”? • Well, yes, but this one can’t be counted (integers) and can’t be represented as any kind of ratio (rational numbers) or conceptualized as a absence/reverse charge of things (negative numbers) or even thought of in terms of place values and approximations (irrational numbers) • This one is more like Santa Claus. Invented for a purpose but still doesn’t make sense in the real world when you think about it much.

So what is an imaginary number? • There’s only one: i = √(-1) • Thought we couldn’t take the square root of a negative? • We can’t in our usual number system; we have to think about things in a new dimension. • If you need additional help, check out the “Class Links” portion of my website.

Okay, how’s this work? • Treat i like any other variable; if it’s multiplied or divided by a number, they are their own “term” (think about when combining like terms). • You handle almost everything the same way you would with any other variable or symbol (like π). • The only exception: exponents.

Exponents on i • If it’s even, divide the exponent by two; odd numbers become negatives, even numbers become positive (because they are two -1’s multiplied together). • If it’s odd, subtract one from the exponent, do the above thing, and then stick an i on it. • e.g., i3 = -i • i4 = 1

Examples • √(-16) = • √(16*-1) = • √(16)* √(-1) = • 4*i • √(-49) =

Examples • We can add or subtract like terms: • (6 + 3i) – (4 – 11i) = • 6 + 3i – 4 + 11i = • 6 – 4 + 3i + 11i = • 2 + 14i

Examples • We can FOIL too: • (2 – 3i)(1 + 4i) = • 2*1 + 2*4i + (-3i*1) + (-3i*4i) = • 2 + 8i + (-3i) + (-12i2) = • 2 + 5i + (-12*-1) = • 2 + 5i + 12 = • 14 + 5i

Imaginary numbers