Question

How many significant figures do these numbers have? a) \(5.44 \times 10^{8}\) b) \(1.008 \times 10^{-5}\) c) 43.09 d) 0.0000001381

Short answer

a) 3; b) 4; c) 4; d) 4

Step-by-step solution

Understanding Significant Figures

Significant figures are the digits in a number that carry meaningful contributions to its measurement accuracy. This includes all non-zero numbers, zeroes between non-zero digits, and trailing zeroes in the decimal portion.

Analyzing 5.44

The number \(5.44 \times 10^{8}\) has three significant figures. Each digit (5, 4, and 4) is non-zero, contributing to the count.

Analyzing 1.008

The number \(1.008 \times 10^{-5}\) contains four significant figures. This includes the non-zero digits (1 and 8) and the zeroes between them, which are also significant.

Analyzing 43.09

The number 43.09 has four significant figures. All non-zero numbers are significant, and any zero between non-zero digits is also significant.

Analyzing 0.0000001381

The number 0.0000001381 has four significant figures. Leading zeroes, those before the first non-zero digit, aren't counted. The significant figures start at 1, and continue through 3, 8, and 1.

Key concepts

Scientific Notation

Scientific notation is a method of writing numbers that are too large or too small to be conveniently written in decimal form. This notation helps express numbers by using powers of ten. For instance, instead of writing really big or small numbers like 0.00000001 or 1,000,000,000, we use scientific notation to simplify them. In scientific notation, a number is written as the product of two numbers:

• A coefficient, which is a number greater than or equal to 1 but less than 10.
• A power of 10, which can be positive or negative, indicating how many places the decimal point has been moved.

For example, the number 0.00045 can be written as \(4.5 \times 10^{-4}\), and 6,500,000 can be expressed as \(6.5 \times 10^{6}\). Scientific notation is especially useful in dealing with the significant figures of a number, as it clearly shows which digits are important in terms of accuracy and precision.

Measurement Accuracy

Measurement accuracy refers to how close a measured value is to the true value. It's an important concept in science and mathematics, where precision is required. The accuracy of a measurement is directly related to the number of significant figures in that measurement. More significant figures usually indicate a more accurate measurement. However, the measuring device's capability also affects accuracy. A ruler with millimeter markings, for example, allows for more precise measurements than one with only centimeter markings. When expressing measurements in scientific notation, significant figures ensure that the number reflects the measurement's level of accuracy. This is crucial when communicating measurements clearly in scientific contexts.

Non-Zero Digits

Non-zero digits are the backbone of significant figures. Any non-zero digit in a numerical value is automatically considered significant because it directly contributes to the measurement's precision. For example, in the number 5,573, each digit is non-zero and hence, each is significant. This is straightforward with numbers written in standard form as well as in scientific notation. Non-zero digits, along with certain kinds of zeroes, form the basis of calculating how many significant figures a number has. When counting significant figures, these non-zero digits play a crucial role, ensuring that the number is as accurate and precise as needed for scientific and mathematical work.

Trailing Zeroes

Trailing zeroes are the zeroes that appear after the last non-zero digit in a number. Whether these zeroes are considered significant depends largely on where the decimal point is placed.

• If there is a decimal point present, then trailing zeroes are significant. For instance, in the number 45.00, the zeroes are significant because they indicate a measurement precision to the hundredth place.
• If there is no decimal point, trailing zeroes are not considered significant. Therefore, in the number 1500 without a decimal point, the zeroes are not significant.

Trailing zeroes help determine the exactness of a measurement. By recognizing which zeroes are significant, one can understand how precisely a value was measured or calculated. This distinction becomes particularly important when numbers are written in scientific notation, as it clearly denotes how many digits are significant.