Question

State the number of significant digits in each of the following: (a) \(0.05 \mathrm{~cm}\) (b) \(0.50 \mathrm{~cm}\) (c) \(25.0 \mathrm{~cm}\) (d) \(20.50 \mathrm{~cm}\)

Short answer

(a) 1 significant digit; (b) 2 significant digits; (c) 3 significant digits; (d) 4 significant digits.

Step-by-step solution

Introduction to Significant Digits

In any given measurement, significant digits (or figures) refer to the meaningful digits that contribute to its precision. This includes all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion.

Evaluating 0.05 cm

In the measurement 0.05 cm, the leading zeros are not significant, they merely indicate the position of the decimal. Here, only the number 5 is significant. Therefore, there is 1 significant digit.

Evaluating 0.50 cm

In the measurement 0.50 cm, the leading zero is not significant but the trailing zero is significant because it is after a decimal and another significant digit (5). This means there are 2 significant digits.

Evaluating 25.0 cm

In the measurement 25.0 cm, all digits are significant. The digits 2 and 5 are significant, and the trailing zero is also significant because it is after a decimal point and follows another significant figure. This makes a total of 3 significant digits.

Evaluating 20.50 cm

In the measurement 20.50 cm, all zeros are significant. The number has 4 digits overall, which are all significant. Therefore, there are 4 significant digits in total.

Key concepts

Precision in Measurements

Measurements are essential in science and daily life, but accuracy alone is not enough; precision is just as important. Precision in measurements refers to how closely repeated measurements of a value agree with each other and how detailed the measurement is.

• It involves careful consideration of significant digits, which help determine the reliability of a measurement.
• Precision also reflects how finely a measurement can be given, which depends on the instrument used.

It is important to remember that a highly precise measurement might still be inaccurate if the equipment is faulty or the method flawed.
Having precise measurements enables scientists and researchers to make consistent comparisons and draw valid conclusions.

Trailing Zeros

Trailing zeros can be confusing, but they play a crucial role in determining the precision of a measurement. Trailing zeros are zeros that appear at the end of a number after any non-zero digits.

• For example, in the number 0.50, the '0' is a trailing zero.
• In decimal numbers, trailing zeros are significant only if they follow a non-zero number.

This is because they indicate the precision of the number.
In scientific notation, trailing zeros can significantly impact the value's representation. By understanding their importance, one can better grasp measurement reliability.

Non-zero Digits

Non-zero digits are the backbone of significant figures. These are digits from 1 to 9, and they are always significant because they directly contribute to the precision of a measurement.

• Every other rule for determining significant figures builds around non-zero digits.
• Figures like 5.67 or 123 have non-zero digits, each considered significant.

Recognizing non-zero digits is a straightforward way to quickly grasp the start of analyzing any measurement's precision.
They hold a guaranteed place in any counting of significant figures, making them indispensable in evaluations.

Significant Zeros

Zeros can sometimes be misleading when it comes to determining significance, but they are vital once understood properly.

• Significant zeros are those that are either between other significant digits or come after a decimal point following a significant digit.
• In numbers like 20.5, all digits are significant due to the zero being sandwiched between significant figures.

Another example is '2.00'; here zeros indicate a high level of precision because they appear after the decimal and follow a non-zero digit.
Grasping the concept of significant zeros ensures that the precision of any measurement is evaluated correctly, maintaining the accuracy of scientific and everyday calculations.