Problem 4 - Fundamentals of Mathematics

Question

Find the sums. $$ (-36)+(-9) $$

Step-by-Step Solution

Answer

The sum is -45.

Step 1: Identify the Signs

First, we need to identify the signs of the numbers involved in the addition. Here both numbers, -36 and -9, have negative signs.

Step 2: Add the Absolute Values

Since both numbers are negative, we add their absolute values. The absolute value of -36 is 36 and the absolute value of -9 is 9. Adding these gives us 36 + 9 = 45.

Step 3: Apply the Sign Rule

The rule for adding two negative numbers is to keep the negative sign for the resulting sum. Thus, the result of -36 + (-9) is -45.

Key Concepts

Negative Numbers

In mathematics, negative numbers represent values less than zero. They are typically indicated by a minus sign (-) placed before the number. Understanding negative numbers is crucial because they show direction, for example, a debt or a temperature below freezing.
Negative numbers lie to the left of zero on the number line. If you move to the left, the numbers start decreasing from zero to negative infinity. Here are some key things to remember about negative numbers:

• When you add a negative number, you essentially move to the left on the number line.
• Negative numbers are less than zero, meaning they are always to the left of positive numbers and zero on the number line.

Handling negative numbers can seem tricky at first, especially with operations like addition or subtraction, but with practice, they become easier to manage.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering its direction. It is always a non-negative value represented by the symbol | | around the number. For example, the absolute value of -5 is written as |-5|, which equals 5.
To find the absolute value, follow these simple rules:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.
• The absolute value of zero is zero.

When adding or subtracting, finding the absolute value is helpful, especially when dealing with negative numbers. It allows us to see differences or totals without initially worrying about the direction (positive or negative) of the numbers involved.

Addition Rules

When we talk about adding numbers, it’s important to understand how the signs of the numbers affect the result. In this context, the signs can be either positive or negative, and there are specific rules when performing addition:
When adding two numbers with the same sign - positive or negative - you:

• Find the absolute values of the numbers involved to simplify the addition.
• Add the absolute values together.
• Keep the common sign of the numbers.

For the equation (-36) + (-9), both numbers have negative signs. You first look at their absolute values: 36 and 9, respectively. When you add these absolute values (36 + 9 = 45), based on the addition rules, you retain the negative sign, resulting in -45.
Learning these addition rules will help you solve many arithmetic problems with confidence.