Problem 2 - Chemistry for Allied Health

Question

Express the following values in decimal notation. a. \(3.8 \times 10^{-3}\) b. \(9.21 \times 10^{5}\) c. \(7.91 \times 10^{-2}\) d. \(2.5 \times 10^{6}\) e. \(3.42 \times 10^{-8}\) f. \(5.4 \times 10^{5}\) g. \(3 \times 10^{-3}\) h. \(7.34 \times 10^{2}\) i. \(9.8 \times 10^{-4}\) j. \(6 \times 10^{7}\) k. \(4.20 \times 10^{-6}\) l. \(4.20 \times 10^{6}\)

Step-by-Step Solution

Answer

a. 0.0038, b. 921000, c. 0.0791, d. 2500000, e. 0.0000000342, f. 540000, g. 0.003, h. 734, i. 0.00098, j. 60000000, k. 0.0000042, l. 4200000

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is written as the product of a number (usually between 1 and 10) and a power of 10. This exercise requires converting such numbers into standard decimal notation by either moving the decimal point to the left (for negative exponents) or to the right (for positive exponents).

Step 2: Convert a. \(3.8 \times 10^{-3}\) to Decimal Notation

For \(3.8 \times 10^{-3}\), move the decimal 3 places to the left, as the exponent is -3. Therefore, \(3.8\) becomes \(0.0038\).

Step 3: Convert b. \(9.21 \times 10^{5}\) to Decimal Notation

For \(9.21 \times 10^{5}\), move the decimal 5 places to the right, as the exponent is 5. Therefore, \(9.21\) becomes \(921000\).

Step 4: Convert c. \(7.91 \times 10^{-2}\) to Decimal Notation

For \(7.91 \times 10^{-2}\), move the decimal 2 places to the left, resulting in \(0.0791\).

Step 5: Convert d. \(2.5 \times 10^{6}\) to Decimal Notation

For \(2.5 \times 10^{6}\), move the decimal 6 places to the right. Therefore, \(2.5\) becomes \(2500000\).

Step 6: Convert e. \(3.42 \times 10^{-8}\) to Decimal Notation

For \(3.42 \times 10^{-8}\), move the decimal 8 places to the left, resulting in \(0.0000000342\).

Step 7: Convert f. \(5.4 \times 10^{5}\) to Decimal Notation

For \(5.4 \times 10^{5}\), move the decimal 5 places to the right, so \(5.4\) becomes \(540000\).

Step 8: Convert g. \(3 \times 10^{-3}\) to Decimal Notation

For \(3 \times 10^{-3}\), move the decimal 3 places to the left, resulting in \(0.003\).

Step 9: Convert h. \(7.34 \times 10^{2}\) to Decimal Notation

For \(7.34 \times 10^{2}\), move the decimal 2 places to the right, so \(7.34\) becomes \(734\).

Step 10: Convert i. \(9.8 \times 10^{-4}\) to Decimal Notation

For \(9.8 \times 10^{-4}\), move the decimal 4 places to the left, resulting in \(0.00098\).

Step 11: Convert j. \(6 \times 10^{7}\) to Decimal Notation

For \(6 \times 10^{7}\), move the decimal 7 places to the right, so \(6\) becomes \(60000000\).

Step 12: Convert k. \(4.20 \times 10^{-6}\) to Decimal Notation

For \(4.20 \times 10^{-6}\), move the decimal 6 places to the left, resulting in \(0.0000042\).

Step 13: Convert l. \(4.20 \times 10^{6}\) to Decimal Notation

For \(4.20 \times 10^{6}\), move the decimal 6 places to the right, so \(4.20\) becomes \(4200000\).

Key Concepts

Decimal Notation

Decimal notation is the regular number format that we use every day. It is the way of writing numbers based on the decimal system, which uses base 10. In decimal notation, each digit represents a quantity based on its position relative to the decimal point.
For example:

• In the number 345.67, the digit '3' is in the hundreds place, '4' in the tens place, and '5' in the units place.
• '6' is in the tenths place, and '7' is in the hundredths place.

Decimal notation is straightforward and helps in simplifying the representation of numbers. However, when dealing with very large or small numbers, it can become cumbersome. That's where scientific notation comes in handy, allowing us to express these numbers more concisely by using exponents.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. They are used as a shorthand in mathematics to indicate how many times a number, called the base, is multiplied by itself. For example, in the expression \( 10^3 \), 10 is the base and 3 is the exponent, indicating that 10 is multiplied by itself three times: \( 10 \times 10 \times 10 = 1000 \).
Exponents are central to scientific notation, which expresses numbers as a product of a number (usually between 1 and 10) and a power of 10.

• For a positive exponent, the decimal moves to the right.
• For a negative exponent, the decimal moves to the left.

Understanding exponents is crucial as they allow us to write both very large and very small numbers in a compact form, making calculations and comparisons much simpler.

Large Numbers

Large numbers can sometimes be difficult to comprehend and write out completely. They often have many zeros, making them cumbersome to handle in calculations. For example, in cases like \( 6 \times 10^{7} \), using scientific notation helps simplify:

• For \( 6 \times 10^{7} \), the number is expressed as 60,000,000 in decimal notation, indicating seven places to the right after 6.

Scientists and mathematicians use this method because it simplifies the process of calculation and understanding. Large numbers frequently arise in fields like astronomy or biology, where distances and quantities can be enormous. By converting to decimal notation from scientific notation, like \( 2.5 \times 10^{6} \) into 2,500,000, we can easily interpret and manipulate these numbers without the bulkiness and potential error that comes from writing out large numbers manually.

Small Numbers

Small numbers can be equally challenging to handle as they often involve many zeros that precede the significant digits. Scientific notation provides a practical solution to this by allowing very small numbers to be expressed more succinctly, which is especially useful in disciplines like chemistry or physics.

For instance:

• The expression \( 3.42 \times 10^{-8} \) translates to 0.0000000342 in decimal notation.
• Similarly, \( 9.8 \times 10^{-4} \) becomes 0.00098.

This conversion is done by moving the decimal point to the left to represent that the number is smaller than one. Using scientific notation helps prevent mistakes and makes it easier to communicate such minute values without getting bogged down in a sea of zeros. This clarity is essential for accurately conveying the scale of measurements and amounts.