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A decimal number is a way to represent numbers that include a fraction of a whole, using a decimal point to separate the whole number part from the fractional part. For example, the number 20.19 has a '20' as the whole number and '.19' as the fraction.

Decimal numbers are based on the base-10 number system, which means each place to the right of the decimal point represents a power of one-tenth. The first place is the tenths, the second is the hundredths, and so on. In our example, '1' is in the tenths place and '9' is in the hundredths place, making the fractional part '19' represent nineteen hundredths.

When we talk about the number of decimal places, we are counting how many digits appear after the decimal point. This is important because it indicates the precision of the number. More decimal places mean a more precise value. However, if a number has no decimal point, like the number 193, it is considered to have no decimal places.

In contrast to decimal numbers, an integer is a whole number that does not have a fractional part or decimal point. It can be positive, negative, or zero. Examples of integers include -2, 0, 7, and 42. It's essential to understand that every integer can become a decimal number by adding a decimal point and a trailing zero. For instance, the integer 7 can also be written as 7.0.

Regarding the exercise, since we determined that integers don't have numbers after a decimal point, they are said to have zero decimal places. This characteristic is an integral part of mathematical notation and helps distinguish between exact values and approximate ones that are represented with decimals.

Mathematical notation comprises the symbols and numbers used to represent mathematical objects, expressions, and the structure of equations. It is a concise and efficient language that allows mathematicians and students to communicate complex ideas unambiguously.

Understanding mathematical notation requires recognizing different kinds of numbers and symbols, such as the decimal point, which indicates a separation between the integer and fractional parts of a number. Each symbol and position conveys specific information; for example, in the previous sections, we saw how the absence of a decimal point signifies an integer with zero decimal places. This structured way of writing also extends to operations, functions, relationships, and more within mathematics.