Question

Find the sum. $$-4+10+(-6)$$

Short answer

The sum of -4, 10, and -6 is 0.

Step-by-step solution

Understand the problem

The problem presents three numbers, -4, 10 and -6. The task is to simply add these together.

Add the first two numbers

First, add the -4 and 10 together. Remember that when adding a positive number to a negative one, the numbers are effectively being subtracted. So, -4 + 10 equals 6.

Add the resulting sum to the third number

Then, add the sum of the first two numbers, 6, to the third number, -6. This equals 0.

Key concepts

Arithmetic Operations

Arithmetic operations are the foundation of mathematics that equip us to handle numbers in various ways, including addition, subtraction, multiplication, and division. When adding integers, we combine the values to arrive at a sum.

Imagine you have a series of steps you need to walk through; arithmetic operations are like the signs guiding you how to proceed. For instance, when confronted with a problem such as \( -4+10+(-6) \), we must first identify the operation required — in this case, it's addition. We then follow the specific rules that govern how to combine these integers.

Remember, when we add a positive number to a negative one, we're essentially finding the difference between their absolute values — think of it as the distance between two points on a line. If the positive number is larger, the sum will be positive, and if the negative number is larger, the sum will be negative. Their absolute values offset each other until one outweighs the other, resulting in the sum.

Negative Numbers

Negative numbers are a way to express values less than zero and are represented by a minus sign (-). They are essential in understanding subtraction and can be viewed as opposite in direction to positive numbers.

When you think of negative numbers in the context of adding integers, imagine you are moving in a different direction on a number line. If positive numbers move you to the right, negative numbers move you to the left. Now, let's translate this to our example: when we have \( -4 + 10 \), we start at -4 on the number line and move 10 steps to the right, landing us at 6. However, when we add \( -6 \), we must take 6 steps back to the left, which returns us to our starting point, or 0.

The proficiency in dealing with negative numbers helps evade confusion and makes it straightforward to comprehend more complex mathematical concepts that involve negative values.

Algebraic Expressions

Algebraic expressions are phrases that can contain numbers, operations, and variables. They can be as simple as \( 2 + 3 \) or as complex as \( 4x - 5y + 2z \). In our example, although there are no variables, \( -4 + 10 + (-6) \) is considered an algebraic expression as it involves arithmetic operations with integers.

One can look at algebraic expressions as sentences that tell a story. For instance, they might represent a situation, like the difference between the amount you earn and the amount you spend. Breaking down these expressions into manageable parts helps us process and solve complex real-world problems.

In the problem given, we're essentially told to add a debt of 4 (\( -4 \)), a credit of 10 (\( +10 \)), and another debt of 6 (\( -6 \)). Understanding algebraic expressions teaches us to analyze and solve these real-world scenarios, and, like any language, the more practice we get, the more comfortable we become with its rules and nuances.