Problem 10 - Fundamentals of Mathematics

Question

Find the sums. $$ -3+5 $$

Step-by-Step Solution

Answer

The sum is +2.

Step 1: Understand the Problem

We need to find the sum of two numbers: \(-3\) and \(+5\). This involves addition of a negative number with a positive number.

Step 2: Identify the Rule for Addition of Numbers with Different Signs

When adding numbers with different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.

Step 3: Apply the Rule

The absolute value of \(-3\) is 3, and the absolute value of \(+5\) is 5. Subtract the smaller absolute value from the larger: \(5 - 3 = 2\). The number with the larger absolute value is \(+5\), so the result is positive.

Step 4: Write the Final Answer

The sum of \(-3\) and \(+5\) is \(+2\).

Key Concepts

Negative Numbers

Negative numbers are a fundamental concept in mathematics and are essentially numbers less than zero. They are denoted by a minus sign (−) placed in front of the number. For example, in the exercise, we have the negative number -3. Here are some essential points about negative numbers:

• Negative numbers represent values less than zero and can be thought of as the opposite of positive values.
• When combined in operations like addition or subtraction, negative numbers can change the result significantly.
• In terms of position on a number line, negative numbers are located to the left of zero.
• Understanding how negative numbers interact with other numbers is crucial for solving problems involving temperature, debt, or altitude changes.

Negative numbers might seem tricky at first, but with practice, they become easier to work with.

Positive Numbers

Positive numbers are numbers greater than zero and are often considered the standard type of numbers we use daily. These numbers don't usually have a sign, but they can be explicitly denoted with a plus sign (+), like +5 in the exercise. Here’s what you should know about positive numbers:

• Positive numbers indicate quantities that are more than nothing—for instance, money in your account or objects you possess.
• On a number line, positive numbers lie to the right of zero.
• They are the building blocks of basic arithmetic and are used to perform operations like addition, multiplication, and division straightforwardly.
• Their simplicity helps clarify concepts when learning mathematics, making them a crucial part of number operations.

Remembering that positive numbers are greater than zero is key to understanding problems involving different types of numbers.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It's denoted by two vertical lines, like this: | -3 | = 3 and | +5 | = 5. Here are some core aspects of absolute value:

• The absolute value of both -3 and +3 is 3, because they are both three units from zero.
• It makes no difference whether the number is positive or negative; all that matters is the distance from zero.
• This concept is particularly useful in problems like the one presented, where you need to decide which number has the greater impact based on their absolute values.
• Absolute values are helpful in real-world contexts, such as measuring temperature changes or financial gains/losses.

Comprehending absolute value can significantly improve your ability to solve a wide range of math problems, especially those involving sums of numbers with different signs.