Problem 68 - Elementary and Intermediate Algebra

Question

Perform the operations. Subtract 512 from \(-47\)

Step-by-Step Solution

Answer

The result is -559.

Step 1: Understand the Problem

We need to subtract 512 from -47. This means we are calculating \(-47 - 512\). Here, both numbers are negative, and we are subtracting a larger number from a smaller one.

Step 2: Rewrite the Expression

Rewriting the subtraction of a positive number as adding its negative allows us to see the operation more clearly: \(-47 - 512 = -47 + (-512)\).

Step 3: Add the Numbers

When you add two negative numbers, add their absolute values and attach a negative sign to the result. Calculate the sum of the absolute values: \(47 + 512 = 559\). Thus, \(-47 + (-512) = -559\).

Step 4: Write the Result

The result of subtracting 512 from -47 is -559.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are typically used to represent values like debts, temperatures below freezing, or other deficits. On the number line, negative numbers are located to the left of zero.

• When dealing with negative numbers, it's important to remember their direction on the number line.
• In operations involving negative numbers, such as addition and subtraction, signs play a crucial role.
• Knowing how to handle negative numbers is essential for solving many mathematical problems.

Visualize negative numbers as going backward or down, in a sense, they are the opposite of positive numbers.

Subtraction

Subtraction is one of the basic arithmetic operations. It involves taking away a number from another. This operation is denoted by the minus sign (-).

• Subtraction can be thought of as the inverse of addition.
• When subtracting a larger number from a smaller number, the result is negative.
• Changing subtraction into the addition of a negative number can be particularly useful when dealing with integers.

For example, in our problem, we convert \(-47 - 512\) into \(-47 + (-512)\), simplifying the calculation.

Absolute Value

The absolute value of a number is the distance from zero on the number line, regardless of direction. It is always a non-negative number.

• Represented by two vertical bars, like \(|-47|\) or \(|512|\).
• For real numbers, the absolute value of a negative number is its positive counterpart.
• Understanding absolute value is crucial for operations like addition and subtraction of integers.

For instance, finding the absolute values of \(-47\) and \(512\) helps us calculate their sum when both numbers are negative.

Addition of Integers

Adding integers can be straightforward when both numbers have the same sign but requires a bit more attention when their signs differ.

• When adding two positive numbers, the result is positive.
• When adding two negative numbers, add their absolute values and give the result a negative sign.
• For a positive and a negative number, find the difference between their absolute values and keep the sign of the larger number.

In the exercise, \(-47 + (-512)\), we focus on adding like-signed integers, getting \(-559\) by adding their absolute values and keeping the negative sign.