Problem 2 - Technical Mathematics with Calculus

Question

Adding and Subtracting Signed Numbers Combine as indicated. $$274+(-412)$$

Step-by-Step Solution

Answer

-138

Step 1: Understanding the Numbers

Identify that 274 is a positive number and -412 is a negative number. When adding a negative number to a positive number, the operation turns into subtraction.

Step 2: Performing the Subtraction

Subtract the absolute value of the negative number from the positive number: 274 - 412.

Step 3: Calculate the Difference

Calculate the difference between the absolute values. Since the absolute value of 412 is greater than 274, the result will be negative.

Step 4: Find the Result

Subtract the numbers: 274 - 412 = -138. The result is a negative number because the number with the larger absolute value (412) is negative.

Key Concepts

Understanding Positive and Negative Numbers

When we talk about numbers, each one comes with a sign: either positive or negative. Think of a number line stretching in both directions; positive numbers go to the right and negative numbers to the left of zero. Positive numbers represent quantities we have, while negative numbers often represent a lack or deficit of something.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering its direction. To visualize, if you have the number -5, it is 5 units away from zero. Similarly, the number +5 is also 5 units away from zero. Both numbers have an absolute value of 5. In mathematical terms, the absolute value of a number is always non-negative.

Subtraction of Integers

Subtracting integers can be trickier than adding them because you must pay attention to their signs. To subtract a negative number is the same as adding its positive counterpart. So if you have to subtract -3 from 5, you're actually adding 5 + 3, resulting in 8. This switch is due to the concept that subtracting a deficit is akin to gaining that amount.

Arithmetic Operations

Arithmetic operations with signed numbers follow specific rules. With addition, like signs result in adding the absolute values and keeping the shared sign. With unlike signs, subtract the smaller absolute value from the larger one and assign the sign of the number with the larger absolute value. This is why in the example 274 + (-412), we end up with a negative result. The absolute value of -412 is larger than that of 274, and since -412 is negative, so is our result.