Which number in each of the following pairs is smaller? a. 4.9 x 10⁻³ or 5.5 x 10⁻⁹

Verified step by step guidance

1. Identify the numbers in scientific notation: 4.9 \times 10^{-3} and 5.5 \times 10^{-9}.
2. Compare the exponents of 10 in both numbers. The number with the smaller exponent is smaller.
3. Here, -3 is greater than -9, so 5.5 \times 10^{-9} has the smaller exponent.
4. Since 5.5 \times 10^{-9} has the smaller exponent, it is the smaller number.
5. Conclude that 5.5 \times 10^{-9} is smaller than 4.9 \times 10^{-3}.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing numbers as a product of a coefficient and a power of ten. It simplifies the representation of very large or very small numbers, making them easier to compare and calculate. For example, 4.9 x 10⁻³ means 4.9 divided by 1000, while 5.5 x 10⁻⁹ means 5.5 divided by 1,000,000,000.

Comparing Exponents

When comparing numbers in scientific notation, the exponent of ten plays a crucial role. A larger negative exponent indicates a smaller number. For instance, 10⁻³ (0.001) is larger than 10⁻⁹ (0.0000000055), which means that any number multiplied by 10⁻³ will be larger than the same number multiplied by 10⁻⁹.

Decimal Representation

Understanding how to convert scientific notation into decimal form is essential for comparison. The number 4.9 x 10⁻³ converts to 0.0049, while 5.5 x 10⁻⁹ converts to 0.0000000055. By comparing these decimal values directly, one can easily determine which number is smaller.