Question

Express each of the following in decimal notation: a. \(149 \times 10^{6} \mathrm{~km}\) (average distance between the Earth and the Sun) b. \(7.9 \times 10^{-11} \mathrm{~m}\) (radius of a hydrogen atom) c. \(4.54 \times 10^{9}\) yr (age of Earth) d. \(6.4 \times 10^{6} \mathrm{~m}\) (radius of Earth)

Short answer

a. 149,000,000 km, b. 0.000000000079 m, c. 4,540,000,000 yr, d. 6,400,000 m

Step-by-step solution

- Convert scientific notation to decimal (Part a)

To convert the given scientific notation to decimal for part a, move the decimal point 6 places to the right since the order of magnitude is positive and has exponent 6. Fill any empty places with zeros.

- Convert scientific notation to decimal (Part b)

For part b, since the exponent of 10 is -11, move the decimal point 11 places to the left. Since we're moving to the left, start the number with zeros followed by the original number.

- Convert scientific notation to decimal (Part c)

Part c has a positive exponent of 9. Move the decimal 9 places to the right. Fill in any empty places with zeros.

- Convert scientific notation to decimal (Part d)

For part d, the exponent is 6, which is positive. Therefore, move the decimal point 6 places to the right. Fill the empty places with zeros if necessary.

Key concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It works by combining a decimal base (usually a number between 1 and 10) with an exponent of 10. The exponent indicates how many places the decimal point would move to convert the number into its full decimal form. For example, the number 149 million, which is a long number in decimal form, can be written more succinctly as \(149 \times 10^{6}\).

Understanding scientific notation requires familiarity with exponents, as they are fundamental to representing the magnitude of the number. In fact, scientific notation is often used in science and engineering because it can simplify calculations and allow us to easily communicate quantities that would otherwise take many digits to write out fully.

Decimal Notation

Decimal notation is the standard form we use to represent numbers on an everyday basis. It employs the base 10 system and includes a decimal point to indicate fractions. To convert a number from scientific to decimal notation, you adjust the decimal point based on the exponent of 10. Positive exponents mean we move the decimal point to the right, creating a larger number, while negative exponents dictate moving it to the left, resulting in a smaller number.

For example, the scientific notation \(7.9 \times 10^{-11}\) becomes a decimal number with the decimal point moved 11 places to the left, creating a very small number. This precise movement of the decimal point is what changes the scientific notation to the more familiar decimal form.

Order of Magnitude

The order of magnitude gives us a relative idea of the size of a number. It's determined by the exponent in the scientific notation of a number. Essentially, the order of magnitude tells us how many powers of 10 there are in the number. For instance, the age of Earth being \(4.54 \times 10^{9}\) years means it has an order of magnitude of 9, indicative of a large scale of time.

When comparing numbers, the order of magnitude can be very useful in determining which numbers are of a similar size, or how much larger or smaller one is than the other, without requiring an actual calculation of the full decimal values. It's a way to quickly grasp the scale and size of values that are involved in a problem.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. In scientific notation, exponents are used to show how many times the number 10 is to be used in a multiplication to get the original number. If you have \(10^{6}\), it means you multiply 10 by itself 6 times.

A positive exponent, such as 6 in \(6.4 \times 10^{6}\), indicates a larger number, while a negative exponent, like -11 in \(7.9 \times 10^{-11}\), indicates a smaller one. Understanding how to manipulate the exponents is crucial to converting between scientific notation and decimal notation effectively.