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Scientific notation is a method used by scientists and engineers to express very large or very small numbers in a compact form. This notation takes the form of a single digit, from 1 to 9, followed by a decimal point and additional significant digits, multiplied by a power of ten. For example, the number 500 can be written as \(5 \times 10^2\) in scientific notation.

When converting from scientific notation to decimal notation, the power of ten indicates how many places to move the decimal point. If the exponent is positive, move the decimal point to the right; if it is negative, move it to the left. This simplicity is what makes scientific notation so useful for calculations involving very large or very small numbers.

Exponents are a critical part of scientific notation and provide a way to represent repeated multiplication of a number by itself. The exponent, located above and to the right of the base number, tells you how many times to multiply the base by itself. For example, \(2^3\) means \(2 \times 2 \times 2\), which equals 8.

An important rule to remember is that a negative exponent indicates division by the base number that many times, instead of multiplication. So \(10^{-3}\), which appears often in scientific and engineering notation, means \(1/(10 \times 10 \times 10)\) or \(0.001\). Being comfortable with exponents is key to skillfully converting between different numerical notations.

Decimal notation is the standard form we use for writing numbers. It involves the use of a decimal point to separate the whole number part from the fractional part of the number. This is the notation most people are familiar with, and it does not involve any exponents.

When converting to decimal notation from scientific or engineering notation, such as in the exercise with the number \(18.640 \times 10^{-3}\), the process simply involves moving the decimal point the number of spaces indicated by the exponent. In this case, moving the decimal point three places to the left converts it to \(0.018640\), making it readily understandable in its decimal form. This process of conversion is essential for making numbers manageable and easier to read in everyday contexts.