A directed number is a number with a sign (direction): Positive and negative. The direction (or sign) of the number adds meaning to what is happening, depending on what the number is describing.
For example, in financial situations or bookkeeping, positive money would be income (receiving money) while negative is money going out (expenses, you spent money on something).
You want the net of all money flows to be greater than zero (positive, which means that means that you will have more money than what you started with). If the net money flow is negative, then you spent more than you earned and your bank account will be decreasing.
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There are many examples of daily life applications of real numbers. Some of these examples include clocks and calendars.
I would be greatly surprised if you will find any examples that ude real numbers - whether or not it is in real life!