Can you help me understand how to only divide by i? Example: 14+i/i

2 Answers
Jun 6, 2018

Dividing by i is the same as multiplying by -i.

Explanation:

Assuming you mean \frac{14+i}{i}, otherwise 14+i/i = 14+1=15

But anyway, since you ask it, let's see how to divide by i in general: you can perform some sort of rationalization, multiplying and dividing by i:

\frac{14+i}{i}*\frac{i}{i} = \frac{i(14+i)}{i^2}= \frac{i(14+i)}{-1} = -i(14+i)

So, dividing by i is the same as multiplying by -i.

This makes particularly sense if you notice the periodicity of the powers of i:

i^0 = 1
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i
...

In general, you have i^{4n+k} = i^k. But this is true for negative exponents as well!

i^{-4} = 1
i^{-3} = i
i^{-2} = -1
i^{-1} = -i
i^0 = i
...

In fact, you can write

1/i = i^-1 = i^{4*(-1)+3} = i^3 = -i

And this is why dividing by i is like multiplying by -i

Jun 6, 2018

You multiply by 1 in the from i/i; this makes the divisor become -1 because i xx i = -1

Explanation:

Given: (14+i)/i

Multiply by 1 in the form of i/i

(14+i)/i i/i

The denominator becomes -1:

(i(14+i))/-1

Use the distributive property to multiply each term in the denominator by i

(14i+i^2)/-1

We know that i^2 = -1:

(14i-1)/-1

Divide by -1:

1 -14i