Problem 38 - Fundamentals of Mathematics

Question

For the following 4 problems, perform the indica ted operations $$ -26+7-52 $$

Step-by-Step Solution

Answer

The result of the operation is -71.

Step 1: Simplify the Expression

Start by adding the first two numbers together: \(-26 + 7\).To do this, subtract 7 from 26, because 26 is a larger number with a negative sign: \(-26 + 7 = -19\).

Step 2: Perform the Next Operation

Now take the result from Step 1, which is \(-19\), and subtract 52 from it: \(-19 - 52\).Since both numbers are negative, add their absolute values and keep the negative sign: \(-19 - 52 = -(19 + 52) = -71\).

Key Concepts

Addition of Integers

Adding integers may seem challenging at first, especially when dealing with negative numbers. However, with some understanding, it becomes straightforward. When you add two positive integers, you simply combine their values. For example, if you add 6 and 3, you get \[ 6 + 3 = 9 \].

• Positive plus Positive = Larger Positive

When you add a positive integer and a negative integer, it is similar to subtraction. You focus on the difference between these numbers. For example, consider adding -4 and 3: here, you subtract 4 from 3 (or the smaller absolute value from the larger one) and use the sign of the larger absolute value. Thus, \[ -4 + 3 = -1 \].

• Negative plus Positive = Subtract and use the sign of the larger absolute value

Adding two negative integers means you combine them like adding two positive values, but the result will have a negative sign. For instance, when adding -2 and -5, add the absolute values and then apply the negative sign: \[ -2 + (-5) = -(2 + 5) = -7 \].

• Negative plus Negative = Larger Negative

Subtraction of Integers

Subtracting integers involves changing the operation to addition and reversing the sign of the number being subtracted.
This is sometimes called "adding the opposite." So instead of directly subtracting, we add the opposite of the integer we want to subtract.
For example, instead of doing 7 - 3, we think of it as adding the opposite: \[ 7 + (-3) = 4 \].

• Change the problem to addition
• Use the opposite sign of the subtracted integer

If both numbers involved are negative, use similar thinking. For example, \[ -5 - (-2) \] becomes \[ -5 + 2 \]. Here, you add the negative 5 to positive 2, resulting in \[ -3 \].

• Subtracting a negative is like adding a positive

Subtraction can help consolidate integer addition skills since each subtraction scenario can convert into an addition once you understand the change in signs.

Negative Numbers in Mathematics

Negative numbers are a fundamental concept in mathematics. They reside on the left side of zero on the number line.
While positive numbers denote quantities that increase, negative numbers represent values that decrease.
For example, -3 is three units less than zero.

• Located on the left of the number line
• Indicate a reduction or opposite of positive value

Negative numbers are used in various real-world contexts like representing debt, temperatures below freezing, and elevation below sea level. Understanding how to operate with negative numbers is vital for handling more complex mathematical problems later on.
When adding, subtracting, multiplying, or dividing with negative numbers, it’s crucial to remember the rules regarding their signs, which significantly affect the outcome.
For any calculation involving negative numbers, a good practice is to visualize or sketch a number line to better see the relative positions and changes.