Question

Which number in each of the following pairs is smaller? a. \(4.9 \times 10^{-3} \mathrm{~s}\) or \(5.5 \times 10^{-9} \mathrm{~s}\) b. \(1250 \mathrm{~kg}\) or \(3.4 \times 10^{2} \mathrm{~kg}\) c. \(0.0000004 \mathrm{~m}\) or \(5.0 \times 10^{2} \mathrm{~m}\) d. \(2.50 \times 10^{2} \mathrm{~g}\) or \(4 \times 10^{5} \mathrm{~g}\)

Short answer

a. 5.5 × 10^{-9} s; b. 3.4 × 10^2 kg; c. 4.0 × 10^{-7} m; d. 2.50 × 10^2 g

Step-by-step solution

Compare the two numbers in the first pair

Convert both numbers to a common form or compare their exponents if they are in scientific notation. For the pair 4.9 × 10^{-3} s and 5.5 × 10^{-9} s, observe the exponents: -3 and -9. The number with the smaller exponent (more negative) is smaller. Hence, 5.5 × 10^{-9} s is smaller.

Compare the two numbers in the second pair

The second pair is 1250 kg and 3.4 × 10^2 kg. Convert 1250 kg to scientific notation: 1.25 × 10^3 kg. Now compare the exponents: 3 (from 1.25 × 10^3) and 2 (from 3.4 × 10^2). The number with the smaller exponent (2) is smaller. Therefore, 3.4 × 10^2 kg is smaller.

Compare the two numbers in the third pair

The third pair is 0.0000004 m and 5.0 × 10^2 m. Convert 0.0000004 m to scientific notation: 4.0 × 10^{-7} m. Now, compare the exponents: -7 and 2. The number with the smaller exponent (more negative) is smaller. Thus, 4.0 × 10^{-7} m is smaller.

Compare the two numbers in the fourth pair

The fourth pair is 2.50 × 10^2 g and 4 × 10^5 g. Compare the exponents: 2 and 5. The number with the smaller exponent is smaller. Hence, 2.50 × 10^2 g is smaller.

Key concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers. It involves writing a number as a product of a coefficient and a power of ten. For example, instead of writing 0.0000004, you can write it as 4.0 × 10^{-7}. This makes calculations easier and prevents errors when dealing with many zeros.
In scientific notation, the coefficient is a number between 1 and 10, and the exponent tells you how many times you have to multiply or divide the coefficient by ten. For example, in the number 4.9 × 10^{-3}, 4.9 is the coefficient and −3 is the exponent. This method is especially useful in science and engineering where measurements can vary widely in magnitude.

Exponent Comparison

Comparing numbers in scientific notation can be simplified by focusing on their exponents. The smaller the exponent (more negative), the smaller the number. For instance, in the pair 4.9 × 10^{-3} s and 5.5 × 10^{-9} s, you compare the exponents −3 and −9. Since −9 is smaller than −3, 5.5 × 10^{-9} s is the smaller number.
Sometimes, you need to convert numbers to scientific notation to make this comparison easier. For example, 1250 kg can be written as 1.25 × 10^3 kg. Now, you can easily compare this with 3.4 × 10^2 kg by looking at the exponents (3 and 2, respectively).

Measurement Units

In measurements, units are crucial for understanding quantities. They provide context to the numbers in scientific notation. For instance, seconds (s) measure time, kilograms (kg) measure mass, and meters (m) measure length.
When comparing measurements, ensure that you are considering the same units. In the given exercise, all pairs of numbers are in the same units, so you only need to compare their magnitudes. However, it’s always good practice to double-check the units to avoid mixing up different types of measurements, like comparing kilograms to grams without converting one of them first.

Order of Magnitude

Order of magnitude refers to the scale or size of a number, often represented in scientific notation. It helps in understanding which numbers are significantly larger or smaller than others. The exponent in scientific notation gives a quick sense of this magnitude.
For example, 2.50 × 10^2 g (250 grams) and 4 × 10^5 g (400,000 grams) are several orders of magnitude apart. This means there is a significant difference between these two values. The term 'order of magnitude' is especially useful in fields like astronomy or physics, where comparing vastly different scales is common. Understanding this concept helps in grasping the vastness or minuteness of different values.