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Arithmetic involves the basic operations of mathematics, including addition, subtraction, multiplication, and division. These operations are the foundation upon which more complex mathematical concepts are built. When working with numbers, it's crucial to understand how these operations interact with each other, especially when negative numbers are involved.

For instance, in the arithmetic expression \( -65 - (-65) \), we see the operation of subtraction being applied. It is a common misconception that addition only increases value and subtraction always decreases value. However, when subtracting negative numbers, the operation can actually increase the value or balance it to zero, which brings us to the core concept of subtracting negative numbers.

Subtracting negative numbers often puzzles students, but it can be understood with a simple rule: subtracting a negative number is the same as adding the corresponding positive number. This is often taught as 'two negatives make a positive'.

When confronted with an expression like \( -65 - (-65) \), you are essentially doing two things. First, you identify that the double negative can be simplified. Second, you apply the rule, replacing the subtraction of a negative with addition. So, \( -65 - (-65) \) transforms into \( -65 + 65 \). Suddenly, you're no longer reducing the number's value; you're potentially increasing it or neutralizing it. It's essential to practice this concept to build confidence and understanding.

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. While the previous example \( -65 - (-65) \) is a numeric expression, algebraic expressions extend this idea to include variables. For example, \( -x - (-y) \) would be an algebraic expression.

The key to understanding algebraic expressions lies in recognizing patterns and rules from arithmetic operations and then applying these to expressions with variables. When you see a subtraction sign in front of a variable, remember that you are dealing with negative values. To simplify these expressions, you use the same arithmetic rules. Algebra builds directly on your knowledge of arithmetic, making the transition from dealing with known quantities to unknowns much smoother.