Round each number to three significant figures. (a) \(10,776.522\) (b) \(4.999902 \times 10^{6}\) (c) \(1.3499999995\) (d) \(0.0000344988\)

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. This notation is especially useful in fields of science and engineering where such numbers are common. In scientific notation, a number is represented as the product of a number between 1 and 10 and a power of 10. For example, the number 2,900 is written as 2.9 × 103 in scientific notation.

To convert a number into scientific notation, you should first move the decimal point to the right of the first non-zero digit. The number of places you move the decimal point is the exponent on the 10. If you move the decimal to the left, the exponent is positive; if you move it to the right, the exponent is negative. For instance, the number 0.0054 would be 5.4 × 10-3 in scientific notation.

Rounding numbers is a fundamental mathematical process used to simplify numbers to a desired level of precision. This process becomes incredibly useful when dealing with numbers with many decimal places or when an approximate value is sufficient. To round a number, you'll need to decide how many significant digits you want to keep, which often depends on the context of the problem.

Here's a simple way to round numbers: locate the digit that is at the position you want to round to, observe the digit immediately to its right, and then increase the target digit by one if the digit to the right is five or higher. If the digit to the right is four or lower, leave the target digit the same. After rounding the target digit, replace all digits to the right with zeros if the number is a whole number, or remove them if it's a decimal. For instance, rounding 6.432 to two significant digits would give us 6.4.

Significant digits, also known as significant figures, are the digits in a number that carry meaningful information about its precision. These include all non-zero numbers, any zeros between non-zero numbers, and trailing zeros in a decimal number. Leading zeros, those to the left of the first non-zero digit, are not considered significant. For example, 0.0078 has two significant digits (7 and 8).

When performing mathematical operations, the number of significant digits can affect the precision of the answer. It's a common rule that the result of a calculation cannot be more precise than the least precise number used in the calculation. Hence, knowledge of how to correctly identify and use significant digits when rounding is essential for maintaining the accuracy of measurements and computations.