Question

Express the following ordinary numbers in scientific notation: (a) 80,916,000 (b) 0.000000015 (c) 335,600,000,000,000 (d) 0.000000000000927

Short answer

(a) \(8.0916 \times 10^7\), (b) \(1.5 \times 10^{-8}\), (c) \(3.356 \times 10^{14}\), (d) \(9.27 \times 10^{-13}\).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. A number is written in scientific notation when it is in the form of \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer.

Convert 80,916,000 to Scientific Notation

First, identify the significant figures: 8.0916. This number has five digits, so you'll multiply by 10 raised to the power of the number of places the decimal has moved. Move the decimal point 7 places to the left (from 80,916,000 to 8.0916), giving us: \(8.0916 \times 10^7\).

Convert 0.000000015 to Scientific Notation

Identify the significant figures: 1.5. Move the decimal 8 places to the right to position the decimal point after the first significant figure (from 0.000000015 to 1.5), giving: \(1.5 \times 10^{-8}\).

Convert 335,600,000,000,000 to Scientific Notation

Identify the significant figures: 3.356. Move the decimal 14 places to the left (from 335,600,000,000,000 to 3.356) to convert the number, resulting in: \(3.356 \times 10^{14}\).

Convert 0.000000000000927 to Scientific Notation

Identify the significant figures: 9.27. Move the decimal 13 places to the right (from 0.000000000000927 to 9.27), resulting in: \(9.27 \times 10^{-13}\).

Key concepts

Significant Figures

Significant figures are the digits in a number that carry meaningful information about its precision. They are crucial in scientific notation, as they help us determine how to express a number more efficiently. Let's break down why and how we identify significant figures:

• The non-zero digits are always significant. For example, in 807, "8" and "7" are significant.
• Any zeros between significant digits are significant. For example, in 8.01, all digits are significant.
• Leading zeros are never significant. Thus, in 0.00552, only "552" are counted as significant.
• Trailing zeros in a decimal number are significant since they indicate the precision of the measurement, such as in 2.5300.

When converting to scientific notation, only significant digits should be maintained. This ensures precision and accuracy, like in the example 80,916,000, where the significant figures are 8.0916. We ignored the trailing zeros as they don't add precision.

Decimal Movement

In scientific notation, moving the decimal is a crucial step. It helps us express large or small numbers in a more manageable form. The rule of thumb is to place the decimal after the first non-zero digit, which helps create a number between 1 and 10 for the coefficient in scientific notation.Depending on whether the original number is large or small, the direction and number of places you move the decimal differ:

• If the number is large, move the decimal left. The number of places moved equals the positive exponent of 10.
• If the number is small, move the decimal right. The number of places moved equals the negative exponent of 10.

For instance, to convert 335,600,000,000,000 to scientific notation, move the decimal 14 places left, resulting in: 3.356. Hence, it's expressed as \(3.356 \times 10^{14}\). Similarly, for 0.000000015, move 8 places right, resulting in \(1.5 \times 10^{-8}\). This process simplifies reading and calculating with very large or very small numbers.

Large Numbers Representation

Representing large numbers efficiently is a key aspect of scientific notation. When faced with extremely large numbers, like billions or trillions, scientific notation offers a succinct way of expression that also preserves accuracy.Let's see how we apply this to an ordinary large number:

• Identify and capture all significant figures, ignoring insignificant zeros.
• Move the decimal after the first significant figure.
• Determine the power of 10 by counting decimal places moved.

For 335,600,000,000,000, scientific notation condenses it into a manageable form of \(3.356 \times 10^{14}\). This process highlights only meaningful figures while placing the scale of the number in context with an exponent. By converting to scientific notation, we simplify complex calculations without sacrificing precision. It's an essential tool in mathematics, physics, and many other fields where handling large values efficiently is imperative.