Problem 14 - Fundamentals of Mathematics

Question

Find the sums. $$ 35+(-78) $$

Step-by-Step Solution

Answer

The sum is -43.

Step 1: Understanding the Problem

We need to find the sum of 35 and -78. This involves adding a positive number (35) and a negative number (-78).

Step 2: Set up the Equation

Write the addition problem as an equation: \[ 35 + (-78) \]

Step 3: Use the Rule for Adding Positive and Negative Numbers

When adding a positive number and a negative number, we subtract the smaller absolute value from the larger absolute value, and keep the sign of the larger absolute value. Here, the absolute values are 35 and 78. Subtract them: \[ 78 - 35 = 43 \]

Step 4: Determine the Sign of the Result

Since the number with the larger absolute value is -78 (a negative number), the result will be negative. So, the sum is \[ -43 \]

Step 5: Write the Final Answer

The sum of 35 and -78 is \[ -43 \]

Key Concepts

Understanding Positive and Negative Numbers

When dealing with integers, it's important we understand the difference between positive and negative numbers. Positive numbers are those greater than zero, like 1, 2, and 35. You can think of these numbers as being located on the right side of the number line. They represent values that are above zero, such as credits in a bank account or gains in a game.

Negative numbers, on the other hand, are less than zero. Examples include -1, -10, and -78. These numbers can be found on the left side of the number line and often represent debts, losses, or temperatures below freezing. Both positive and negative numbers are crucial in mathematics because they allow us to measure and express changes in direction, quantity, or position.

Grasping the Concept of Absolute Value

Absolute value is a fundamental concept when working with integers. It refers to the distance of a number from zero on the number line, regardless of direction. The absolute value tells us how far away a number is from zero, without considering whether it's positive or negative.

For example:

• The absolute value of 35 is written as \(|35| = 35\).
• The absolute value of -78 is written as \(|-78| = 78\).

Notice that the absolute value makes both numbers positive because distance can't be negative. This concept is critical when adding or subtracting positive and negative numbers, as it helps us to determine which number is larger when ignoring the sign.

Easily Handling Subtraction of Integers

Subtracting integers, especially when they involve both positive and negative numbers, can seem tricky, but it's manageable with a few rules. If you think of subtraction as "adding the opposite," it becomes simpler.

Here's how you can handle subtraction of integers:

• When subtracting a positive number, think of it as adding its negative. For example, \(8 - 3\) can be thought of as \(8 + (-3)\).
• When subtracting a negative number, you actually add the absolute value of that number. For instance, \(8 - (-3)\) becomes \(8 + 3\).

In the exercise we solved, we added a negative integer (-78) to a positive integer (35). We used the absolute values to calculate the difference: \( 78 - 35 = 43\). Then, because the larger absolute value came from the negative integer (-78), our result was negative \(-43\). Understanding the role of absolute values and recognizing patterns in integer operations can make solving these problems much easier.