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Adding integers is a fundamental skill in mathematics, encompassing the combination of positive and negative whole numbers. When integers with different signs are added, such as in the example (-35) + 47, one can think of it as balancing two opposing forces. To effectively add integers with different signs, it's essential to determine which number has a greater absolute value, because this will indicate the overall sign of the answer.

For our exercise, the process begins with identifying the larger number, 47, and recognizing its positive sign. Next, we calculate the difference between the absolute values of the numbers (ignoring signs initially) to find the true magnitude of the result. Subtracting the smaller absolute value from the larger gives us 12. Finally, because the larger original number was positive, the resulting sum inherits this positive sign, leading to a final answer of +12. This process is consistent for all cases of integer addition with different signs, ensuring students can tackle these problems with confidence.

The absolute value of an integer is its distance from zero on the number line, without considering direction. It's denoted by two vertical bars surrounding the number—for example, the absolute value of -35 is written as \(|-35|\), and its value is 35. This concept is crucial when adding integers with different signs because it helps to determine the true magnitude of each number without the complexity of its sign.

When we work with the exercise (-35) + 47, we initially disregard the signs and focus solely on the absolute values: \(|-35| = 35\) and \(|47| = 47\). By comparing these, we can clearly see that 47 has a greater magnitude. During the addition process, the difference between these magnitudes determines the numerical part of the result, setting the stage for the final answer. Understanding absolute value is not just useful for addition, but also for all operations with integers, as it is the key to grasping the concept of magnitude versus direction in mathematics.

The understanding of positive and negative numbers is essential for integer addition. In our number system, positive numbers represent quantities greater than zero, while negative numbers represent quantities less than zero. Positive numbers are often associated with gains or increases, whereas negative numbers often represent losses or decreases.

When adding numbers like in the exercise (-35) + 47, think of -35 as a debt or decrease and 47 as an asset or increase. The overall outcome depends on which is greater, the 'debt' or the 'asset.' Since here the 'asset' exceeds the 'debt,' the result is a net 'gain,' or a positive number. This analogy is helpful for visualizing the combination of positive and negative numbers, making it easier to understand why a larger positive number results in a positive sum, while a larger negative number would give a negative sum. Grasping this relationship is fundamental to working with all mathematical operations involving positive and negative numbers.