15 days includes the start day and the end day, for example, Sunday, 5 April to Sunday, 19 April is 15 days, but the span is often referred to as 2 weeks. 19-5=14 which is also 2 weeks. A 2-week holiday is usually 15 days, because you travel and return on the same day of the week. It's like fence posts and fencing. If you have 15 fence posts, you only need 14 fence panels. In the same way a week is sometimes referred to as 8 days, meaning the span of days, so there are only 7 days in between. An octave (from oct- meaning 8) in music is 8 notes (do re mi fa so la te do) because we start and end on do, but there are only 7 different degrees: do re mi fa so la te. So there's nothing wrong with the mathematics, it's just a matter of interpretation. 15 days is a span encompassing 14 days or a fortnight; 8 days is a span encompassing 7 days or a week; an octave is a span encompassing 7 notes or a scale; 15 fence posts is a span encompassing 14 panels.