Question

Evaluate each of the following mathematical expressions, and express the answer to the correct number of significant digits. a. \(44.2124+0.81+7.335\) b. \(9.7789+3.3315-2.21\) c. \(0.8891+0.225+4.14\)

Short answer

a. \(52\) b. \(10.9\) c. \(5.25\)

Step-by-step solution

a. \(44.2124+0.81+7.335\)

1. Identify the number with the least significant digits: \(0.81\) has two significant digits. 2. Perform the mathematical operations: \(44.2124 + 0.81 + 7.335 = 52.3574\) 3. Round the result to the same number of significant digits: The rounded result is \(52\).

b. \(9.7789+3.3315-2.21\)

1. Identify the number with the least significant digits: \(2.21\) has three significant digits. 2. Perform the mathematical operations: \(9.7789 + 3.3315 - 2.21 = 10.9004\) 3. Round the result to the same number of significant digits: The rounded result is \(10.9\).

c. \(0.8891+0.225+4.14\)

1. Identify the number with the least significant digits: \(0.225\) has three significant digits. 2. Perform the mathematical operations: \(0.8891 + 0.225 + 4.14 = 5.2541\) 3. Round the result to the same number of significant digits: The rounded result is \(5.25\).

Key concepts

Mathematical Expressions

Mathematical expressions are a way to represent a set of numbers and operations in a concise manner. When you see an expression like \(44.2124 + 0.81 + 7.335\), it is essentially a short form for some actual numbers waiting to be added together. Each number in an expression, known as a 'term,' can affect the outcome based on its value and placement within the expression. Understanding the composition of mathematical expressions allows you to effectively break them down and perform operations like addition or subtraction step-by-step.

Rounding Numbers

Rounding numbers is an essential skill that helps in simplifying long decimal numbers for easier analysis and comparison. The basic idea of rounding is that you approximate a number to a certain degree of precision, usually determined by the context.

To round a number:

• Identify the place value you are rounding to.
• Look at the digit immediately to the right of your chosen place value.
• If this digit is 5 or greater, increase the rounding place by one; otherwise, leave it as is.

Practically, rounding helps in maintaining consistency, especially when working with numbers of different significant figures.

Addition and Subtraction

The basic operations of addition and subtraction are fundamental in mathematics and are the first set of operations you perform when evaluating mathematical expressions. To carry out these operations efficiently:

• Align the numbers correctly according to their decimal points before adding or subtracting.
• Begin with the numbers on the right-most side and work your way left.
• Take note of carrying (in addition) and borrowing (in subtraction) as these steps are crucial for accurate calculations.

These operations form the base through which you derive more complex solutions.

Numerical Accuracy

Ensuring numerical accuracy involves working with numbers and results in a manner that maintains their integrity and precision where required. A key part of numerical accuracy, especially in scientific and financial calculations, is the concept of significant figures.

When you perform operations and have to decide on the number of significant figures to report:

• Always look for the number within your operation with the least significant figures.
• Your final result should reflect the same number of significant figures as this least precise number.
• This approach helps standardize results so that they are both consistent and fairly representative of the precision of the inputs.

Adhering to these rules ensures the reliability of your results in various fields of work.