Problem 64 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-1.067+(-4.4)\)

Step-by-Step Solution

Answer

The result is \(-5.467\).

Step 1: Identify the Operation

We need to add the two decimal numbers: \(-1.067\) and \(-4.4\). Since both numbers are negative, we will add their absolute values and then apply the negative sign to the result.

Step 2: Align the Numbers

Align the decimals vertically to make sure the decimal points are in a straight line. This will help in adding the decimals correctly:\[\begin{array}{r} -1.067 \ -4.400 \\hline\end{array}\]

Step 3: Add the Absolute Values

Add the aligned decimal numbers as you would add whole numbers:\[\begin{array}{r} -1.067 \+ -4.400 \\hline -5.467\end{array}\]

Step 4: Apply the Negative Sign

Since both numbers were originally negative, the sum is also negative. Therefore, the final result is \(-5.467\).

Key Concepts

Negative Numbers

Dealing with negative numbers can be a bit tricky initially, but once you grasp the concept, it becomes much simpler. The number line is an excellent tool for visualizing negative numbers. Positive numbers appear to the right of zero, while negative numbers are to the left. In our case,

• \( -1.067 \) and \( -4.4 \) are both negative.
• Adding two negative numbers means you are moving further to the left on the number line.

To add them, simply combine their absolute values (ignoring the negative sign) and then reapply the negative sign to your result. This approach ensures the calculation accurately reflects the negative sum.

Absolute Value

Absolute value refers to how far a number is from zero on the number line. It is always a positive number, regardless of whether the original number is positive or negative. For example:

• The absolute value of \( -1.067 \) is \( 1.067 \).
• The absolute value of \( -4.4 \) is \( 4.4 \).

Knowing this helps when dealing with addition and subtraction, especially involving negative numbers. By focusing on the absolute values of the numbers in our exercise, we can perform the addition as if they were positive. When the absolute values were added in our exercise, we got \( 5.467 \). Because both original numbers were negative, the sum is also negative: \( -5.467 \). Understanding absolute values helps maintain clarity when calculations involve integers that vary in sign.

Number Alignment

Number alignment is crucial when performing operations with decimals. This is particularly important when adding or subtracting, as misaligned decimal points can lead to incorrect results. For example, when aligning numbers:

• Align each number by the decimal point so that all digits sit in the correct place value column, such as units, tenths, hundredths, etc.
• Zeros can be added to fill in any gaps after the decimal, ensuring all numbers are the same length.

In our exercise,

• The numbers \( -1.067 \) and \( -4.4 \) were aligned vertically.
• The decimal \(-4.4\) is equivalent to \(-4.400\), adding appropriate zeros to ensure neat alignment.

Once the decimals are well aligned, the addition proceeds smoothly, with the result being easy to read and understand. This practice prevents common errors and strengthens arithmetic skills.