Question

Express each number in decimal notation (i.e., express the number without using scientific notation). (a) \(450 \times 10^{-9} \mathrm{~m}\) (wavelength of blue light) (b) \(13.7 \times 10^{9}\) years (approximate age of the universe) (c) \(5 \times 10^{9}\) years (approximate age of Earth) (d) \(5.0 \times 10^{1}\) years (approximate age of this author)

Short answer

a) 0.00000045 m b) 13,700,000,000 years c) 5,000,000,000 years d) 50 years

Step-by-step solution

Convert (a) to Decimal Notation

To convert from scientific notation to decimal notation, move the decimal point 9 places to the left (since the exponent is negative) for the number 450. This gives: 450 becomes 0.00000045. So, the wavelength of blue light in decimal notation is 0.00000045 meters.

Convert (b) to Decimal Notation

To convert from scientific notation to decimal notation for the number 13.7, move the decimal point 9 places to the right (since the exponent is positive), which yields the decimal notation: 13,700,000,000. Therefore, the approximate age of the universe in decimal notation is 13.7 billion years.

Convert (c) to Decimal Notation

For the number 5 and an exponent of 9, move the decimal point 9 places to the right to convert it into decimal notation: 5,000,000,000. Hence, the approximate age of Earth in decimal notation is 5 billion years.

Convert (d) to Decimal Notation

For the number 5.0 with an exponent of 1, simply move the decimal point 1 place to the right. This changes the number to 50. Therefore, the approximate age of the author in decimal notation is 50 years.

Key concepts

Decimal Notation

Decimal notation, or decimal form, is perhaps the most familiar way of representing numbers for everyday use. It's the standard system for denoting integer and non-integer numbers. It is characterized by the use of a decimal point to separate the integer part from the fractional part of a number. For instance, the number 0.5 signifies that there's half an integer, while the number 2.5 indicates two whole integers plus a half. Understanding decimal notation is critical because it's used in various academic fields, commerce, and everyday life to express quantities, measurements, and values.

Regarding the conversion from scientific to decimal notation, the position of the decimal point is adjusted according to the power of ten. When the exponent is negative, the decimal point moves to the left, indicating a value less than one. Conversely, when the exponent is positive, the decimal point moves to the right, signifying a larger number. This system is integral to accurately translating between different magnitudes, whether you're measuring the length of a molecule or the distance to a star.

Scientific Notation

Scientific notation is a method of writing numbers that are too large or small to be conveniently written in decimal form. It is expressed as the product of a number between 1 and 10 and a power of ten. This form allows for easier comparisons and calculations with very large or very small numbers. In scientific notation, the number is normalized such that there's only one non-zero digit to the left of the decimal.

For example, in the case of the wavelength of blue light, represented as \(450 \times 10^{-9}\), it indicates a very small number. Scientific notation simplifies understanding and processing of this number through normalization and exponentiation, which in a more standard decimal form would involve many zeroes and could be more prone to error if written fully.

Exponential Notation

Exponential notation is a powerful mathematical tool used to express repeated multiplication of a number by itself. It's denoted by a base number and an exponent. The base represents the number being multiplied, and the exponent indicates how many times to use the base in the multiplication.

For example, \(10^3\) means \(10 \times 10 \times 10\), which equals 1000. This notation is essential in scientific notation where the base is often 10. The ease of moving the decimal point in relation to the exponent in base 10 simplifies large calculations, which is why base 10 is prevalently used in scientific and engineering work. It also aligns with our decimal numeral system, indicating powers of ten and making the conversion between exponential and decimal notation intuitive.

Mathematical Notation

Mathematical notation encompasses the symbols used to write mathematical equations and expressions. It's a concise language that allows for the efficient communication of mathematical ideas. The notation includes numbers, operators, grouping symbols, and functions, which together allow for a vast range of expressions from simple arithmetic to complex equations.

Each field within mathematics may have its unique set of symbols and use standard notation differently. For instance, in calculus, a prime symbol (\('\)) denotes a derivative, whereas in linear algebra, bold letters often indicate vectors or matrices. Learning and understanding mathematical notation is fundamental to progress in mathematics, and it evolves as new concepts and relationships are discovered.