Problem 64 - Prealgebra

Question

Find each difference. $$33-(-19)$$

Step-by-Step Solution

Answer

The difference is 52.

Step 1: Identifying the Expression

The problem requires us to find the difference: \(33 - (-19)\). We need to simplify this expression by applying the rules of subtracting a negative number from a positive number.

Step 2: Convert Subtraction to Addition

Recall that subtracting a negative number is the same as adding its positive counterpart. Therefore, transform the expression \(33 - (-19)\) to \(33 + 19\).

Step 3: Calculate the Sum

Now, add the two numbers together. Compute \(33 + 19\). First, add the tens: \(30 + 10 = 40\). Then, add the units: \(3 + 9 = 12\). Combine these results to find the total sum: \(40 + 12 = 52\).

Step 4: Conclude the Difference

The calculation shows that \(33 + 19 = 52\). Therefore, the difference \(33 - (-19)\) simplifies to 52.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are widely used in various fields of science and everyday life. For instance, negative numbers are used to denote temperatures below freezing, elevations below sea level, and financial debts. Understanding them is crucial because they introduce new rules and concepts in mathematics.
When dealing with negative numbers, it’s important to remember:

• Negative numbers are to the left of zero on the number line.
• Adding a negative number is the same as subtracting a positive number.
• Subtracting a negative number is equivalent to adding a positive number.

This latter point is particularly useful in solving problems where you subtract a negative number, as it simplifies calculations and aids in understanding arithmetic operations better.

The Concept of Addition

Addition is one of the four basic arithmetic operations, where we combine numbers to find their total. It builds the foundation for more complex mathematical calculations. With addition, it is essential to understand the following:

• The addition operation is commutative, meaning that the order of adding numbers doesn’t change the sum. For example, \(a + b = b + a\).
• Addition is associative, which means when adding multiple numbers, the grouping of these numbers does not affect the sum. Thus, \((a + b) + c = a + (b + c)\).
• Adding zero to any number does not change its value, as zero is the identity element of addition.

In the context of integer subtraction involving negative numbers, the concept of addition plays a critical role. It transforms subtraction into a familiar operation, making it easier to handle complex expressions.

Importance of Mathematics Education

Mathematics education is crucial for developing problem-solving skills and logical thinking. It provides the tools necessary to tackle various real-life situations involving numerical and analytical problems. Encouraging a strong mathematical foundation helps students:

• Understand and apply mathematical concepts effectively.
• Develop critical-thinking skills.
• Prepare for future careers in science, engineering, and technology fields.

Moreover, in a rapidly advancing technological world, having a solid grasp of mathematical principles aids in processing information and data critically. This is increasingly relevant, as we encounter math in everyday scenarios, from budgeting at home to calculating energy use.

Essentials of Arithmetic

Arithmetic involves the fundamental operations of mathematics: addition, subtraction, multiplication, and division. Mastery of arithmetic is vital, as it serves as the building block for more advanced mathematics. Key points about arithmetic include:

• It is the first step young learners take in mathematics, forming the basis of their numeracy skills.
• Accuracy in basic arithmetic ensures a strong foundation for understanding algebra, geometry, and statistics.
• Arithmetic is essential for developing analytical skills used in everyday decision-making.

Understanding how to subtract, add, and execute these operations with negative numbers enhances mathematical fluency. For instance, applying the principle that subtracting a negative turns into addition simplifies the arithmetic process, making operations straightforward.