Express each number in scientific notation. (a) \(0.000000001 \mathrm{~s}\) (time it takes light to travel \(1 \mathrm{ft}\) ) (b) \(0.143 \mathrm{~s}\) (time it takes light to travel around the world) (c) \(0.000000000001 \mathrm{~s}\) (time it takes a chemical bond to undergo one vibration) (d) \(0.000001 \mathrm{~m}\) (approximate size of a dust particle)

Scientific notation is a system that simplifies the writing and handling of large or very small numbers by expressing them as a product of two parts: the coefficient and the power of ten. It's a shorthand which ensures that these numbers are easier to read, compare, and use in calculations.

Here's how you can communicate numbers in this form: Start with the original number and relocate the decimal point until only one non-zero digit remains to its left. This digit becomes the 'coefficient', and it should be between 1 and 10. The amount of positions you shifted the decimal point is your 'exponent'. If you moved the decimal to the right, as you would with a very small number, your exponent is negative. Conversely, shifting it to the left, for a large number, results in a positive exponent. The final form resembles 'a × 10^b', where 'a' is the coefficient and 'b' is the exponent.

Using our previous exercise examples:

• For the time it takes light to travel 1 ft, we would shift the decimal 9 places to the right to transform '0.000000001 s' into '1 × 10^-9 s'.
• To express the time light takes to travel around the world '0.143 s', we move the decimal 1 place to the right to write '1.43 × 10^-1 s'.

Essentially, this method simplifies complex numbers to a consistent and manageable form while retaining all their significant figures.

The concept of 'order of magnitude' refers to the scale of a value, measured in exponents within the scientific notation context. It is used to compare numbers based on powers of ten, providing a quick sense of size or scale. Essentially, it is a way to describe how many times you need to multiply ten to reach a certain number.

For instance, if we examine the time it takes for a chemical bond to vibrate, expressed in scientific notation as '1 × 10^-12 s', the order of magnitude is -12. This reflects that the number is on the scale of one trillionth. Similarly, the approximate size of a dust particle '1 × 10^-6 m' has an order of magnitude of -6, telling us it's on the scale of one millionth.

In practical terms, if two numbers have different orders of magnitude, it tells us that they differ substantially in size. The greater the difference in the exponent, the larger the gap in their respective scales. Order of magnitude gives us an intuitive understanding of the relative size of numbers within scientific contexts and beyond.

The two primary components in scientific notation are exponents and coefficients. As touched upon in previous sections, the coefficient is a number that usually ranges from 1 to just below 10. It preserves the significant digits of the original number and helps tell us the magnitude in a more direct sense than raw digits.

An exponent, also known as the power of ten, indicates how many places the decimal point was moved. It could be positive or negative, and it directly impacts the order of magnitude. Positive exponents show large numbers, while negative exponents signal small numbers.

Understanding coefficients and exponents is crucial as it expresses the precision and scale of numbers. For instance, '1.43' as the coefficient in '1.43 × 10^-1 s' means we are accurate up to two decimal places. The exponent '-1' suggests that the number is ten times smaller than the base unit. Together, these elements help articulate precise scientific measurements and calculations in a simplified and standardized manner.