Question

How many significant figures are in each of the following numbers? (a) \(0.025\) (b) \(22.4\) (c) \(0.0404\) (d) \(5.50 \times 10^{3}\)

Short answer

(a) 2, (b) 3, (c) 3, (d) 3

Step-by-step solution

Identify significant figures in (a)

For the number 0.025, ignore the leading zeros. The remaining digits are 2 and 5. Since these are non-zero, they are significant. Therefore, 0.025 has 2 significant figures.

Identify significant figures in (b)

For the number 22.4, all digits are non-zero. Therefore, each digit in 22.4 is significant. Thus, 22.4 has 3 significant figures.

Identify significant figures in (c)

For the number 0.0404, ignore the leading zeros. The remaining digits are 4, 0, and 4. Since the zeros between non-zero digits are also significant, 0.0404 has 3 significant figures.

Identify significant figures in (d)

For the number 5.50 × 10^3, we focus on the coefficient 5.50. All non-zero digits and the zero between them are significant. Therefore, 5.50 has 3 significant figures.

Key concepts

Leading Zeros

When determining the number of significant figures in a number, leading zeros are an important concept. Leading zeros are the zeros that appear before the first non-zero digit in a number. These zeros do not count as significant figures because they only serve as placeholders and do not affect the value of the number itself.
For example, in the number 0.025, the leading zeros are the first two zeros before the digit 2. These leading zeros are ignored when counting significant figures. Therefore, only the digits 2 and 5 are considered, giving us 2 significant figures. Remember:

• Leading zeros are never significant
• They only indicate the position of the decimal point

Non-zero Digits

Non-zero digits are always significant because they represent meaningful quantities. Every non-zero digit in a number contributes to its precision and accuracy.
Consider the number 22.4: all the digits (2, 2, and 4) are non-zero and thus are significant. This means 22.4 contains 3 significant figures. To sum up:

• Every non-zero digit is significant
• These digits help indicate the precision of the value

In another example, the number 5.50 × 10³ has 3 significant figures: 5, 5, and the trailing zero. Here, the zero is significant because it is between non-zero digits.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a more manageable form. It uses powers of ten to scale the number, which simplifies calculations and aids in clearly identifying significant figures.
For example, the number 5.50 × 10³ consists of a coefficient (5.50) and a power of ten (10³). Only the digits in the coefficient are considered when counting significant figures. Here, 5.50 has 3 significant figures.
This method helps to:

• Clearly indicate precision
• Avoid confusion with leading or trailing zeros

Scientific notation separates the value into a fraction and exponent, making it easier to manage and count significant digits.