Question

Round off each of the following numbers to three significant digits. a. 0.00042557 b. \(4.0235 \times 10^{-5}\) c. 5,991,556 d. 399.85 e. 0.0059998

Short answer

The rounded numbers to three significant digits are: a. 0.000426, b. \(4.02 \times 10^{-5}\), c. 5,990,000, d. 400.0, and e. 0.00600.

Step-by-step solution

Round off 0.00042557 to three significant digits

We begin counting for the significant digits after the first non-zero digit. So, in this case, 4 is the first significant digit. Now, the third significant digit is 5, and since the next digit after that is greater than or equal to 5 (which is 7 in this case), we add 1 to the third digit. So after rounding, we get 0.000426.

Round off \(4.0235 \times 10^{-5}\) to three significant digits

In the scientific notation, 4.0235 is the coefficient and -5 is the exponent. So we need to round the coefficient to three significant digits. Here, the third significant digit is 2, and since the next digit after that is less than 5 (which is 3 in this case), the last significant digit remains the same. Our rounded number is \(4.02 \times 10^{-5}\).

Round off 5,991,556 to three significant digits

Here, the first three significant digits are 5, 9, 9. The next digit after the third is 1, which is less than 5, so the last significant digit remains the same. When we round it off, it becomes 5,990,000.

Round off 399.85 to three significant digits

The first three significant digits are 3, 9, 9. Since the next digit after the third is greater than or equal to 5 (which is 8 in this case), we add 1 to the third digit. After rounding, we get 400.0.

Round off 0.0059998 to three significant digits

We begin counting for the significant digits after the first non-zero digit. So, in this case, 5 is the first significant digit. Now, the third significant digit is 9, and since the next digit after that is greater than or equal to 5 (which is 9 in this case), we add 1 to the third digit. So after rounding, we get 0.00600.

Key concepts

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. It makes calculations easier and more manageable. The format typically involves a number (called the coefficient) and an exponent of ten. For example, in the expression \(4.0235 \times 10^{-5}\), the number 4.0235 is the coefficient, and \(-5\) is the exponent indicating the decimal point moves five places to the left.

• The coefficient should be a number ranging from 1 to less than 10.
• The exponent shows how many places to move the decimal point.

This notation helps simplify operations like multiplication and division while keeping precision. When rounding numbers in scientific notation, you primarily focus on the coefficient. By rounding the coefficient to a desired number of significant figures, as seen in rounding 4.0235 to 4.02, the clarity of calculations and results is maintained.

Rounding Numbers

Rounding numbers is a method used to simplify figures while retaining a value close to the original. It involves altering the number by decreasing its digits. This method is especially handy when dealing with long decimal places or lengthy numbers.

• Identify the digit to be rounded.
• Check the digit immediately following it.
• If this digit is 5 or more, add 1 to the rounding digit. If less than 5, leave the digit as it is.

For instance, when rounding the number 0.00042557 to three significant digits, look closely at the third significant figure, which is 5 in this instance. Because the next digit (7) is more than 5, we increase the third digit, turning 0.00042557 into 0.000426. Similarly, clearly understanding this process ensures accuracy and precision in mathematics.

Significant Digits

Significant digits are the digits in a number that contribute to its accuracy. These include all non-zero numbers, any zeros between significant digits, or trailing zeros in a decimal. Knowing how to count significant digits is crucial, particularly in scientific measurements.

• Start counting with the first non-zero digit you encounter.
• Include zeros only if they fall between significant digits or come after a decimal point.

For example, in the number 399.85 rounded to three significant digits, the important digits are initially 3, 9, and another 9. Since the next figure (8) is greater than or equal to 5, we enhance the last digit, making it 400.0 with three significant digits. This concept not only refines calculations but is also essential in interpreting scientific data accurately.