Question

Adding integers with the same sign is similar to increasing assets or increasing debts. $$-31+(-9)$$

Short answer

Answer: $$-40$$.

Step-by-step solution

Identifying the sign of the result

Since both numbers are negative, the result will also be negative.

Ignoring the signs and finding the absolute value of the sum

Temporarily ignoring the signs, we have: $$31 + 9$$ Now, we simply add these two numbers together: $$31 + 9 = 40$$ The absolute value of the sum is 40.

Combining the sign with the absolute value

Since we determined that the result would be negative in Step 1, we now combine the negative sign with the absolute value of the sum (40) obtained in Step 2: $$-40$$ The final result is $$-40$$.

Key concepts

Understanding Absolute Value

The concept of absolute value is central to understanding integers, especially when dealing with positive and negative numbers. In simple terms, the absolute value of a number is its distance from zero on the number line, regardless of direction. For example, the absolute value of both \( -3 \) and \( 3 \) is \( 3 \) because both are three units away from zero.

The equation for finding the absolute value of a number \( x \) is denoted as \( |x| \) and is defined by:

• If \( x \) is a positive number or zero, \( |x| = x \).
• If \( x \) is a negative number, \( |x| = -x \), which makes the result positive.

When adding negative integers as in \( -31+(-9) \), finding the absolute value is a helpful intermediate step. By temporarily ignoring the negative signs and adding the absolute values \( |31| + |9| = 40 \), you can determine the magnitude before addressing the sign of the result.

Dealing with Negative Numbers

Negative numbers can be challenging, but they are just as essential in mathematics as positive numbers. They represent quantities less than zero and are used to describe loss, debt, or decreases. On the number line, negative numbers lie to the left of zero while positive numbers are to the right.

When adding negative numbers, the combined effect is similar to accumulating debt. If you owe \( \$31 \) and then borrow an additional \( \$9 \), your total debt is the sum of those amounts. To find the sum, you don't need to alter the numbers' negative status initially; simply add their absolute values and then apply the negative sign to the result, which reflects the increased debt - hence a negative sum.

Integers Addition Step by Step

Let's delve into the step-by-step process of adding integers using our given exercise as an example: \( -31+(-9) \).

Step 1: Identify the Sign of the Numbers

First and foremost, identify the sign of each number to be added. Since both \( -31 \) and \( -9 \) are negative, the result of their addition will also be negative.

Step 2: Add the Absolute Values

Next, look at the absolute values: \( |31| + |9| = 40 \). This step focuses solely on the size of the numbers, not their sign.

Step 3: Combine the Sign and the Absolute Value

Finally, combine the sign determined in Step 1 with the sum of the absolute values obtained in Step 2, leading to \( -40 \). This final result confirms that when adding two negative integers, we get a more significant negative integer, indicating an increase in the 'negative' direction on the number line.