Problem 26 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-16+(-10)$$

Step-by-Step Solution

Answer

-26

Step 1: Identify the numbers to be combined

We have two numbers: -16 and -10. Both are negative numbers.

Step 2: Apply the rule for adding negative numbers

When adding two negative numbers, we add their absolute values and keep the negative sign. The absolute value of -16 is 16, and the absolute value of -10 is 10.

Step 3: Combine the absolute values

Add the absolute values: 16 + 10 = 26.

Step 4: Apply the negative sign

Since both numbers are negative, the result also carries a negative sign, so the answer is -26.

Key Concepts

Absolute Value

The absolute value of a number is how far it is from zero on a number line, without considering direction. This is a crucial concept when working with negative numbers. The absolute value is always non-negative. For example, the absolute value of \(-16\) is 16, because it is 16 units away from zero. Similarly, the absolute value of \(-10\) is 10. When you see vertical bars around a number, such as \(|-16|\), it denotes absolute value.

Absolute values are important in integer operations because they help simplify calculations. In the context of adding negative numbers, we add their absolute values to find their total magnitude, then reapply the negative sign. This makes understanding the process more manageable.

Negative Numbers

Negative numbers are numbers less than zero, usually used to represent loss, debt, or temperatures below freezing. They are found to the left of zero on the number line. When you work with negative numbers in addition, a few rules stand out:

• Adding two negative numbers results in a negative number.
• Subtracting a negative number is like adding a positive number.
• The further a negative number is from zero, the smaller its value (e.g., \(-16\) is less than \(-10\)).

When adding negative numbers, like in the example \(-16 + (-10)\), you are combining their effects. To do this efficiently, we first deal with their absolute values to understand their magnitude, then attach the negative sign to the result.

Integer Operations

Integer operations encompass addition, subtraction, multiplication, and division involving whole numbers and their opposites (negative numbers). These foundational operations are governed by rules that help simplify computations and accurate calculus.

One key rule for addition involving integers is:

• Add two positive integers normally.
• Add two negative integers by summing their absolute values and keeping a negative sign.
• For a positive and a negative integer, subtract the smaller absolute value from the larger and take the sign of the larger integer.

This rule assists in performing calculations effortlessly, helping to avoid mistakes, especially when multiple integers are involved. Mastering integer operations paves the way for understanding more advanced mathematical concepts.