Problem 20 - Fundamentals of Mathematics

Question

Use a calculator to find each sum. $$ 673+(-721) $$

Step-by-Step Solution

Answer

The sum is -48.

Step 1: Identify the Numbers

We need to find the sum of two numbers: 673 and -721.

Step 2: Understand the Operation

Since one number is positive and the other is negative, this operation is essentially subtraction. We will subtract the smaller absolute value (673) from the larger absolute value (721).

Step 3: Calculate the Subtraction

Subtract 673 from 721: \[ 721 - 673 = 48 \]

Step 4: Determine the Sign of the Result

Since the larger absolute value was from the negative number, the result of the addition will also be negative. Thus, the sum is -48.

Step 5: Confirm Using a Calculator

Use a calculator to confirm: Input 673 + (-721). The calculator should display -48.

Key Concepts

absolute value

The absolute value of a number is a measure of its distance from zero on the number line, without considering its direction. This means it is always a non-negative number. For instance:

• The absolute value of 673 is 673 because it is 673 units away from zero.
• The absolute value of -721 is 721 because it is also 721 units from zero.

Absolute value is useful when comparing numbers, especially when dealing with operations involving positive and negative integers. It helps simplify decisions about which number is "larger" in terms of magnitude. When subtracting or adding positive and negative numbers, always consider their absolute values to determine the correct operation and result.

positive and negative numbers

Positive numbers are numbers greater than zero and are usually written without a sign, such as 673. Negative numbers are less than zero, and they are represented with a minus sign (-), such as -721.
These numbers are opposite in terms of value and placement on the number line. Positive numbers are to the right of zero, while negative numbers are to the left.
When adding these numbers, the sign plays a crucial role. For example, when you add a positive and a negative number, you need to calculate the difference between their absolute values. The number with the larger absolute value determines the sign of the result. In the exercise, since -721 has a larger absolute value than 673, the result of the addition is negative.

subtraction as addition

Subtraction can be thought of as the addition of a negative number. This concept is especially handy when dealing with integer operations. To subtract a number is to add its additive inverse.
In the given exercise of 673 + (-721), instead of performing subtraction explicitly, you use the addition of a negative number to achieve the same result. Essentially:

• Subtracting 721 from 673 translates into adding -721 to 673.

This approach helps in unifying the operations and often simplifies calculations, as demonstrated by converting subtraction into an addition problem. The technique of changing subtraction to the addition of a negative is a key mathematical strategy that aids in solving a variety of problems effectively.