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I don't understand how one can get from:

$1+r_{t+1}$=$\frac{1+i_{t}}{1+\pi_{t+1}}$

to

$1+r_{t+1}$$\approx1+i_{t}-\pi_{t+1}$

by applying the following rule of thumb:

$Z$=$\frac{X\cdot{Y}}{Q}$

implies

$\frac{\Delta{Z}}{Z}$=$\frac{\Delta{X}}{X}$+$\frac{\Delta{Y}}{Y}$-$\frac{\Delta{Q}}{Q}$

Note:

$r_{t+1}$: real interest rate

$i_{t}$: nominal interest rate

$\pi_{t+1}$: inflation rate

I hope someone will help :)

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Argument 1:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$

Take the natural log of both sides.

$$ \log(1+r_{t+1}) = \log(1+i_{t}) - \log(1+\pi_{t+1})$$

Observe that $\log(1 + x) \approx x$ for $x$ near zero because that's the first order taylor expansion. Hence:

$$ r_{t+1} \approx i_{t} - \pi_{t+1}$$

Argument 2:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$ $$ 1+r_{t+1} + \pi_{t+1} + r_{t+1} \pi_{t+1} = 1+i_{t}$$

And if $r_{t+1} \pi_{t+1}$ is closer to zero

$$ r_{t+1} + \pi_{t+1} \approx i_{t}$$

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