Significant Figures: Rules and Examples

Significant Figures

• Indicate the precision of a number.
• Counted from the first non-zero digit.

Rules for Identifying Significant Figures

• All non-zero digits are significant.
• Zeros between non-zero digits are significant (e.g., 6007 has five significant figures).
• Zeros at the end of an integer may be significant, depending on the required accuracy.
• Zeros at the end of a decimal are significant, as they indicate the level of accuracy (e.g., 0.005020 has four significant figures).
• Zeros before the first non-zero digit are not significant (e.g., 0.005020 has four significant figures).

Determining Significant Figures: Decimal Example

• Consider the decimal 0.00501400.
• Non-zero digits (5, 1, 4) are significant.
• Zeros between non-zero digits are significant.
• Zeros at the end of the decimal are significant.
• Zeros before the first non-zero digit are not significant.
• 0.00501400 has six significant figures.

Determining Significant Figures: Integer Example

• Consider the integer 803,000.
• Non-zero digits (8, 3) are significant.
• Zeros between non-zero digits are significant.
• Zeros at the end of an integer are significant depending on accuracy.
• 803,000 can have three, four, five, or six significant figures, depending on the accuracy specified.

Rounding Off Numbers

• Identify the digit to be rounded off based on the desired number of significant figures.
• Look at the digit to the right of the digit being rounded off.
• If the digit to the right is less than 5, the digit being rounded off remains unchanged, and the digits to the right are replaced with zeros.
• If the digit to the right is 5 or greater, add 1 to the digit being rounded off, and the digits to the right are replaced with zeros.
• Rounding off 63,479 to two significant figures results in 63,000.
• Rounding off 2,476 to two significant figures results in 2,500.
• Rounding off 68.79 to three significant figures results in 68.8.
• Rounding off 0.008146 to three significant figures results in 0.00815.

Standard Form

• Expresses numbers as A x 10^n, where 1 ≤ A < 10 and n is an integer.
• Used to write very large or very small numbers.

Changing a Single Number to Standard Form

• Place the decimal point after the first non-zero digit.
• Determine the power of 10 needed to make the equation correct.
• 28 becomes 2.8 x 10^1.
• 0.003025 becomes 3.025 x 10^-3.

Changing a Number in Standard Form to a Single Number

• Move the decimal point according to the power of 10.
• If the exponent is positive, move the decimal point to the right.
• If the exponent is negative, move the decimal point to the left.
• Fill in spaces with zeros as needed.
• 4.17 x 10^5 equals 417,000.
• 8.063 x 10^-5 equals 0.00008063.

Operations of Addition and Subtraction in Standard Form

• Ensure that the numbers have the same power of 10 before performing addition or subtraction.
• If the powers of 10 are different, adjust one of the numbers to match the power of 10 of the other.
• 2.73 x 10^3 + 5.92 x 10^3 = (2.73 + 5.92) x 10^3 = 8.65 x 10^3.
• 7.02 x 10^4 + 2.17 x 10^5 = 0.702 x 10^5 + 2.17 x 10^5 = 2.872 x 10^5.

Operations of Multiplication and Division in Standard Form

• For multiplication: (a x 10^m) x (b x 10^n) = (a x b) x 10^(m+n)
• For division: (a x 10^m) / (b x 10^n) = (a / b) x 10^(m-n)
• (3 x 10^5) x (4.9 x 10^2) = (3 x 4.9) x 10^(5+2) = 14.7 x 10^7 = 1.47 x 10^8
• (6.8 x 10^-3) / (4 x 10^-6) = (6.8 / 4) x 10^(-3 - (-6)) = 1.7 x 10^3

Solving Problems Involving Numbers in Standard Form

• Diameter of Earth: 1.2742 x 10^4 km
• To find the surface area, use the formula: Surface Area = 4πr^2
• Radius (r) = Diameter / 2 = (1.2742 x 10^4 km) / 2 = 6.371 x 10^3 km
• Surface Area = 4 x 3.142 x (6.371 x 10^3)^2 ≈ 5.101 x 10^8 km^2 (correct to four significant figures)