Problem 22 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ 53+(-37) $$

Step-by-Step Solution

Answer

The sum of 53 and -37 is 16.

Step 1: Identify the Operation

Recognize that the problem asks you to add the numbers 53 and -37. This involves adding a positive number (53) and a negative number (-37).

Step 2: Understand Number Relation

When adding a positive number (53) to a negative number (-37), you are essentially finding the difference between the two numbers. The outcome will be positive if the positive number is larger, and negative if the negative number's absolute value is larger.

Step 3: Calculate the Difference

Since 53 is greater than the absolute value of -37, subtract 37 from 53. The calculation: 53 - 37 = 16.

Step 4: Determine Final Sign

After calculating the difference, the sign of the result is the same as the sign of the larger (absolute value) number. Since 53 is larger than 37, the result is positive.

Key Concepts

Integer Addition

Adding integers involves working with both positive and negative numbers. It's crucial to understand how to manage the signs during the process. There are three main scenarios:

• Adding two positive integers: The result is straightforwardly the sum of both numbers and will always be positive. For example, \(5 + 8 = 13\).
• Adding two negative integers: Here, you add the absolute values and then apply a negative sign to the result since both operands are negative. For instance, \((-4) + (-3) = -7\).
• Adding a positive integer and a negative integer: This is like finding the difference between their absolute values. You take the larger absolute value and subtract the smaller one. The sign of the result takes after the sign of the larger absolute value. Such as in the given example: \(53 + (-37) = 16\).

Understanding these basics makes integer addition clear across all possible scenarios.

Number Line

A number line is a visual tool that helps to understand the relative positions of different numbers. It's particularly useful for integer operations:

• Positive numbers are located to the right of zero. As you move further to the right, the numbers increase.
• Negative numbers sit to the left of zero. They decrease further as you move to the left.
• When adding a positive number to a negative one, start at the negative number and move right on the number line by the amount of the positive number.
• If the starting number is positive and the number added is negative, you go left from the starting point.

Using a number line aids in conceptualizing how positive and negative values interact. For instance, starting at \(-37\) and moving 53 steps right gets you to 16.

Absolute Value

The absolute value of a number is its distance from zero on the number line, considered without regard to direction. It's always a non-negative value:

• For a positive number, the absolute value is the number itself. For example, the absolute value of \(53\) is \(53\) itself.
• For a negative number, it becomes positive. For example, the absolute value of \(-37\) is \(37\).

Absolute value is used in integer addition to determine the effect of positive and negative numbers. When adding \(53 + (-37)\), it's essential first to find the absolute values of both numbers, identify the larger one, and then subtract the smaller one. As seen, you calculate \(53 - 37 = 16\). Thus, absolute value helps simplify and resolve the differences in integer addition.

Arithmetic Operations

Arithmetic operations are the core of all mathematics and include addition, subtraction, multiplication, and division. For integer addition involving positive and negative numbers, the operations break down as:

• Recognizing the kinds of numbers involved (positive and negative) allows you to determine which arithmetic operations to undertake. Adding a negative is practically the same as subtracting its absolute value.
• Understanding the rules for combining signs: a positive number plus a negative (or vice versa) involves subtraction of their absolute values.
• After calculating the arithmetic operation, consider the signs involved. If you subtract a smaller absolute value from a larger one, the result's sign is that of the number with the larger absolute value.

For the example of \(53 + (-37)\), you subtract \(37\) from \(53\), understanding that you're essentially subtracting. The operation gives \(16\), a positive number, as \(53\) is larger than \(37\) in terms of absolute value.