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Decimal degree notation is a way of expressing angles using decimal numbers instead of traditional degrees, minutes, and seconds. This method is used in navigation, mapping, and many other fields because it's simpler to work with, especially for calculations. Instead of writing an angle like this: \( 9^{\bullet} 43^{\bullet} \), we convert it into a single decimal number. For example, in the problem given, \( 9^{\bullet} 43^{'} \) is converted to \( 9.72^{\bullet} \). This makes computations and comparisons straightforward. To convert minutes and seconds into decimal degrees, we simply divide by 60 and 3600 respectively. For instance, \( 43^{'} \) becomes \( \frac{43}{60} \) in degrees.

Angles measure the rotation between two rays originating from a common point called the vertex. We use angles in various fields like trigonometry, geometry, astronomy, and even everyday navigation systems. Typically, angles are represented in degrees, with a full circle being \( 360^{\bullet} \). For more precise measurements, an angle can be broken down further into minutes (') and seconds (''), where \( 1^{\bullet} = 60^{'} \) and \( 1^{'} = 60{''} \). This traditional notation can be cumbersome, which is why decimal degrees are often preferred in modern applications. They provide a cleaner and more convenient system, reducing the risk of conversion errors.

Converting degrees and minutes into decimal degrees is straightforward once you understand the conversion factors. First, you keep the whole number part of the degrees. Then you convert the minutes into a fraction of a degree by dividing by 60. For example, to convert \( 43^{'} \) into degrees, you compute \(43 \times \frac{1^{\bullet}}{60} \), which gives \( 0.7167 \) degrees. You then add this to the whole number part of the degrees. In the example given in the exercise, you have \( 9^{\bullet} + 0.7167 = 9.7167 \) degrees. Finally, you round the result to the desired decimal places. Here, rounding \( 9.7167 \) to two decimal places gives \( 9.72^{\bullet} \). This method ensures that your angle measurements are accurate and easy to use for further calculations.