Question

Round each of the following numbers to four significant figures and express the result in standard exponential notation: (a) \(102.53070\), (b) \(656.980\), (c) \(0.008543210\), (d) \(0.000257870\), (e) \(-0.0357202\).

Short answer

The short answers in standard exponential notation are: (a) \(1.025 \times 10^2\), (b) \(6.570 \times 10^2\), (c) \(8.543 \times 10^{-3}\), (d) \(2.579 \times 10^{-4}\), and (e) \(-3.572 \times 10^{-2}\).

Step-by-step solution

(a) Rounding 102.53070 to four significant figures

First, we identify the first four significant figures: 102.5. The fifth significant figure is 3, which is less than 5, so we don't need to round up. Thus, the rounded number is 102.5. In standard exponential notation, this is \(1.025 \times 10^2\).

(b) Rounding 656.980 to four significant figures

First, we identify the first four significant figures: 656.9. The fifth significant figure is 8, which is greater than 5, so we need to round up. Thus, the rounded number is 657.0. In standard exponential notation, this is \(6.570 \times 10^2\).

(c) Rounding 0.008543210 to four significant figures

First, we identify the first four significant figures: 0.008543. The fifth significant figure is 2, which is less than 5, so we don't need to round up. Thus, the rounded number is 0.008543. In standard exponential notation, this is \(8.543 \times 10^{-3}\).

(d) Rounding 0.000257870 to four significant figures

First, we identify the first four significant figures: 0.0002578. The fifth significant figure is 7, which is greater than 5, so we need to round up. Thus, the rounded number is 0.0002579. In standard exponential notation, this is \(2.579 \times 10^{-4}\).

(e) Rounding -0.0357202 to four significant figures

First, we identify the first four significant figures: -0.03572. The fifth significant figure is 2, which is less than 5, so we don't need to round up. Thus, the rounded number is -0.03572. In standard exponential notation, this is \(-3.572 \times 10^{-2}\).

Key concepts

Scientific Notation

Scientific notation is a method to express very large or very small numbers in a concise form. It is particularly useful in fields like chemistry and physics where such numbers frequently occur. This notation involves representing a number as a product of a coefficient (between 1 and 10) and a power of ten. For example, 6500 becomes \(6.5 \times 10^3\).

The main advantage of scientific notation is its ability to simplify arithmetic operations, making it easier to read and manage calculations without losing accuracy. When applied to the exercise solutions:

• 102.5 in scientific notation is \(1.025 \times 10^2\).
• Similarly, 657.0 becomes \(6.570 \times 10^2\).
• Small numbers like 0.008543 are expressed as \(8.543 \times 10^{-3}\).
• And 0.0002579 turns into \(2.579 \times 10^{-4}\).

Understanding this method is crucial for mastering calculations in scientific fields.

Rounding Numbers

Rounding numbers involves reducing the digits in a number while keeping its value similar to the original. This is done by following specific rules to either round up or round down based on the digits that are being dropped off.

Consider a situation where you are asked to round a number to a certain number of significant figures. Identifying these figures is crucial:

• In 102.53070, the rounding depends on the fifth digit, which is less than 5, so the number is rounded to 102.5.
• For 656.980, the fifth digit is greater than 5, requiring an upward rounding to 657.0.
• The process is similar for numbers with leading zeroes; for example, 0.008543210 rounded to four significant figures becomes 0.008543.

Rounding is essential in chemistry where precision is necessary but values need to be manageable.

Chemistry Mathematics

Chemistry mathematics often requires handling calculations involving constant conversions, molar relationships, and dealing with both very large and small quantities. Using concepts like rounding and scientific notation helps simplify these problems.

In chemistry, significant figures are especially important. They indicate how precise a measurement is and dictate the number of meaningful digits in calculations. For example:

• Measurements like concentrations and molecular weights often use such precision.
• Working with measurements requires understanding which decimal places are trustworthy for reliable results.

Mastery of these mathematical skills is key in solving complex chemistry problems accurately.

Numerical Precision

Numerical precision refers to maintaining accuracy in computational results. It is crucial for achieving reliable and accurate outcomes in scientific calculations. Precision is especially critical when the numbers can significantly impact the results.

Identifying and maintaining significant figures helps ensure that the numbers used in calculations represent real-world measurements accurately:

• For instance, in our exercise, keeping four significant figures consistently provides a balance between accuracy and simplicity.
• When dealing with trailling zeroes in a decimal number like 0.008543210, precision mandates retaining the necessary significant figures only.

This precision is vital in chemical calculations, impacting the validity and trustworthiness of results.