Question

The following numbers are in scientific notation. Write them in ordinary notation. a. \(7.050 \times 10^{3} \mathrm{g}\) b. \(4.00005 \times 10^{7} \mathrm{mg}\) c. \(2.3500 \times 10^{4} \mathrm{mL}\)

Short answer

a. 7050 g, b. 40,000,500 mg, c. 23,500 mL

Step-by-step solution

Understand scientific notation

Scientific notation is a way to express very large or very small numbers. A number in scientific notation is written as the product of a number (between 1 and 10) and a power of 10. For example, in the number \(7.050 \times 10^{3}\), the '7.050' is the coefficient and '10^{3}' represents multiplying by 1000.

Convert \(7.050 \times 10^{3} \mathrm{g}\) to ordinary notation

To convert \(7.050 \times 10^{3}\) to ordinary notation, move the decimal point 3 places to the right because the exponent is 3.\(7.050 \times 10^{3} = 7050\) grams.

Convert \(4.00005 \times 10^{7} \mathrm{mg}\) to ordinary notation

To convert \(4.00005 \times 10^{7}\) to ordinary notation, move the decimal point 7 places to the right because the exponent is 7.\(4.00005 \times 10^{7} = 40,000,500.0\) milligrams.

Convert \(2.3500 \times 10^{4} \mathrm{mL}\) to ordinary notation

To convert \(2.3500 \times 10^{4}\) to ordinary notation, move the decimal point 4 places to the right because the exponent is 4.\(2.3500 \times 10^{4} = 23,500\) milliliters.

Key concepts

Understanding Ordinary Notation

Ordinary notation is the standard way we write numbers every day. For example, the number 7050 is in ordinary notation. It represents the value without any exponents or powers of ten. In ordinary notation, the value can be easily read and understood without any conversion. This is useful for everyday counting and simple math problems. However, it can become cumbersome when dealing with very large or very small numbers, which is where scientific notation comes in handy.

Coefficients in Scientific Notation

In scientific notation, a number is expressed as a coefficient multiplied by a power of ten. The coefficient is a number between 1 and 10. For example, in the expression \(7.050 \times 10^{3}\), the number 7.050 is the coefficient. The coefficient determines the significant figures in the number. The coefficient must be chosen such that it always falls within the range of 1 to 10, ensuring a standard way to represent the number before it is scaled by the power of ten.

Understanding Decimal Point Movement

To convert a number from scientific notation to ordinary notation, you need to move the decimal point. The direction and number of places you move the decimal depends on the exponent. For example:
\(7.050 \times 10^{3}\): Move the decimal point 3 places to the right (\(7050\))
\(4.00005 \times 10^{7}\): Move the decimal point 7 places to the right (\(40,000,500.0\))
\(2.3500 \times 10^{4}\): Move the decimal point 4 places to the right (\(23,500\))
This process utilizes the exponent to determine how many places you need to shift the decimal point. Positive exponents move the decimal point to the right, making the number larger, while negative exponents move it to the left, making the number smaller.

Interpreting Exponents in Scientific Notation

Exponents in scientific notation tell us how many places to move the decimal point in the coefficient. A positive exponent means you move the decimal point to the right. For example, in \(7.050 \times 10^{3}\), the exponent 3 tells us to move the decimal point 3 places to the right, converting it to 7050. Conversely, a negative exponent means moving the decimal point to the left. This method allows us to handle extremely large or small numbers conveniently and consistently.