Question

Round off the following to three significant digits: (a) \(9.123 \times 10^{5}\) (b) \(9.456 \times 10^{10}\) (c) \(9.000 \times 10^{-7}\) (d) \(9.075 \times 10^{-12}\)

Short answer

(a) \(9.12 \times 10^5\), (b) \(9.46 \times 10^{10}\), (c) \(9.00 \times 10^{-7}\), (d) \(9.08 \times 10^{-12}\).

Step-by-step solution

Understand the Significance of Digits

To round off a number to three significant digits, we must retain the first three non-zero digits starting from the left. These are considered significant, while any others are rounded based on the value of the subsequent number.

Analyze and Round (a)

For the number \(9.123 \times 10^5\), the first three digits are 912. Looking at the fourth digit, 3, since it is less than 5, we do not round up the third digit. The result is \(9.12 \times 10^5\).

Analyze and Round (b)

Looking at \(9.456 \times 10^{10}\), the first three digits are 945. The fourth digit is 6, which is more than 5, so we round up the last digit of the first three. Therefore, \(9.46 \times 10^{10}\) becomes \(9.46 \times 10^{10}\).

Analyze and Round (c)

For \(9.000 \times 10^{-7}\), the significant digits are 900. There are no digits beyond the first three non-zero numbers, so no rounding is necessary. Thus, it remains \(9.00 \times 10^{-7}\).

Analyze and Round (d)

For \(9.075 \times 10^{-12}\), the first three digits are 907. The fourth digit is 5, which requires rounding up the third digit. Therefore, \(9.08 \times 10^{-12}\) is the result.

Key concepts

Rounding Numbers

Rounding numbers is a fundamental skill, particularly when dealing with significant figures. In scientific and engineering work, precision is crucial, but often, calculations do not require all digits. Imagine you have measured a length of wood as 9.123 meters, but your measurement tool isn't that precise. You might round this to a simpler, more manageable number while retaining accuracy for important digits. In this context, let's learn how to round numbers effectively.

When rounding a number to a specific number of significant digits, you focus on the first few non-zero numbers. If you're instructed to round to three significant figures, you start with the first significant one and look at the next digit to decide.

• If the next digit is 5 or higher, you round the last retained significant digit up by one.
• If the next digit is less than 5, you leave the last retained digit as it is.

For example, when rounding 9.456 to three significant figures: "9, 4, and 5", the next digit is 6. Since it's 5 or higher, you round 5 up to 6, resulting in 9.46. Mastering rounding will make readings less cumbersome without losing meaning.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers conveniently. In scientific fields, you often encounter numbers like 0.0000001 or 1000000, which can be cumbersome to write out repeatedly. Instead, scientific notation simplifies them using powers of ten.

A number in scientific notation is written as the product of a number between 1 and 10 and a power of ten. For example, 9.123 can be written as:

• \(9.123 \times 10^{5}\)
• The exponent 5 indicates how many places to move the decimal point to convert it back to its original whole number form, i.e., 912300.

This representation is not only compact but also allows easier computation, especially with operations like multiplication and division.

To convert a number to scientific notation, identify the significant digits, place the decimal after the first significant digit, and count how many places you've moved the decimal. The power of ten reflects this movement. This method helps keep calculations tidy and reduces the margin for error in scientific work.

Non-Zero Digits

Understanding non-zero digits is essential for working with significant figures. They are the digits in a number that are not zero, and they play a crucial role in determining which parts of a number are meaningful and which are merely placeholders.

When calculating with significant figures, start counting from the first non-zero digit. This counting helps establish the scale of a measurement or a calculated result. Supposing we look at the number 0.004567, the first four zeros are not significant because they only set the decimal's position. The significant figures here are "4, 5, 6, and 7", making these the digits you consider when performing rounding.

• In a number like 5600, if the context suggests that the trailing zeros are significant, you might express it in scientific notation to clarify, like \(5.600 \times 10^{3}\) which indicates four significant digits.

Correctly identifying non-zero digits and understanding their significance ensures your calculations' accuracy and reliability, especially in precise fields like science and engineering. This skill sharpens your mathematical operations and helps communicate values more effectively.