Express these numbers in scientific notation: (a) 0.749 ,( b) \(802.6,\) (c) 0.000000621 .

Scientific notation is a concise way to express very large or very small numbers. It's based on powers of 10, which reflect how many places the decimal point has been moved. To express a decimal in scientific notation, we identify the most significant digit and place the decimal point immediately after it, creating a number between 1 and 10. Then, we count the number of places the decimal has moved from its original position. This count becomes the exponent on 10, which we write alongside the new decimal.

For instance, let's take 0.749, a number less than 1. It's already in the form we need (between 1 and 10) so we can simply write it as \(7.49\) and determine the exponent for 10. In this case, since the decimal hasn't moved, the exponent is 0, yielding \(7.49 \times 10^{0}\) in scientific notation.

Remember, moving the decimal to the right will give a negative exponent, and moving it to the left gives a positive exponent. This is because we are either dividing or multiplying by 10 respectively to get back to the original number.

The exponent in scientific notation is fundamental, as it indicates the actual scale or size of the number. Positive exponents show how many times we multiply by 10 for numbers greater than one, while negative exponents indicate the division by 10 for numbers less than one.

For example, for the number 802.6, the decimal is moved two places to the left to become \(8.026\). Therefore, we've multiplied by 10 twice, indicated by an exponent of 2, written as \(8.026 \times 10^{2}\). In contrast, with a very small number like 0.000000621, we need to move the decimal seven places to the right, resulting in a notation of \(6.21 \times 10^{-7}\). This tells us that to return to the original number, we'd divide by 10 seven times.

The presence of negative or positive exponents quickly signals to us whether the number is a small decimal or a large quantity without having to count zeroes, making comparison and calculation with very big or very small numbers much more manageable.

Converting a standard decimal number to scientific notation is a straightforward process. First, determine the non-zero digit furthest to the left and place a decimal point after it. Next, count the number of places you moved the decimal point; this number will be your exponent, positive if you moved the decimal to the left and negative if to the right.

To convert the number 802.6, for instance, place the decimal after the first significant digit: \(8.026\). Since we've moved the decimal two places, the exponent is 2, resulting in \(8.026 \times 10^{2}\). Likewise, to convert 0.000000621, you would place the decimal to get \(6.21\), and since the decimal moved seven places to the right, the exponent is -7, resulting in \(6.21 \times 10^{-7}\).

It's important to maintain the integrity of the original number during this conversion, so always check your scientific notation by multiplying the decimal by \(10\) to the power of your exponent. This will take you back to the original number if your conversion is correct. Doing this effectively can enhance numerical literacy, especially when dealing with extreme values in science and engineering disciplines.