Problem 45 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ 0-(-10) $$

Step-by-Step Solution

Answer

Answer: The result of subtracting -10 from 0 is 10.

Step 1: Understand the problem

We need to perform the operation:$$ 0-(-10) $$ For that, we need to remember that subtracting a negative number is the same as adding a positive number.

Step 2: Change the subtraction of a negative number into addition

Given the general rule that subtracting a negative number is the same as adding a positive number, we rewrite the expression as: $$ 0+10 $$

Step 3: Perform the addition

Now, all that remains is to perform the addition:$$ 0+10=10 $$Therefore, the result of the operation is 10.

Key Concepts

Integer Operations

Integer operations involve the manipulation of whole numbers, which include both positive and negative numbers. In mathematics, integers are numbers without fractional parts. Operations with integers mostly include addition, subtraction, multiplication, and division. Each of these operations has specific rules that must be followed. Understanding these rules is crucial, especially when working with negative numbers, as it can avoid common pitfalls.

• Addition of Integers: When you add two integers with the same sign, you retain the common sign and add their absolute values. For instance, \(-3 + (-5) = -8\).
• Subtraction of Integers: Subtracting an integer is equivalent to adding its opposite. So, if you need to subtract, say \(3 - 7\), you can think of it as \(3 + (-7)\).
• Multiplication and Division: These are similar in that if both integers have the same sign, the result is positive. If they have different signs, the result is negative.

Understanding integer operations is about recognizing these patterns and applying the correct arithmetic rules in solving mathematical problems.

Addition and Subtraction

Addition and subtraction are fundamental operations in arithmetic, and understanding them is critical, especially when working with negative numbers.

• Addition Rules: When adding integers, it's important to focus on the signs. If both numbers are positive, you simply add their absolute values. The same applies when they are both negative.
• Subtraction Rules: Subtraction can often be a bit trickier. The key rule here is to convert subtraction into the addition of integers. For example, when subtracting a negative number like \(0 - (-10)\), it helps to realize you are essentially adding the positive equivalent of that number, which turns into \(0 + 10\).

This simple conversion can help greatly simplify complex arithmetic operations, allowing you to leverage addition skills rather than getting tangled in subtraction complexities.
Ultimately, breaking down subtraction into addition paves a way to handle operations with ease and less confusion.

Arithmetic Rules

Arithmetic rules are a cornerstone of mathematical operations. These rules guide us in correctly handling different operations involving numbers. They especially apply when working with integers, especially with negative numbers.

• Rule for Subtraction: The pivotal rule concerning subtraction is that subtracting a negative number amounts to adding its positive counterpart.
• Order of Operations: It's always important to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).This ensures operations are performed correctly and in the right sequence.
• Commutative and Associative Properties: These properties aid us when grouping numbers in addition and multiplication. For addition, it means numbers can be in any order and still yield the same sum (e.g., \(2 + 3 = 3 + 2\)).

These arithmetic rules are crucial for making sense of equations and simplifying various mathematical problems. They offer a consistent framework by which all numbers are treated the same way, ensuring accuracy in mathematical operations.