Adding Subtracting Multiplying And Dividing Integers

Mastering the Four Operations with Integers: A Comprehensive Guide

Understanding how to add, subtract, multiply, and divide integers is fundamental to success in mathematics. Integers encompass all whole numbers, both positive and negative, and zero. While seemingly simple, mastering these operations forms the bedrock for more complex mathematical concepts. This comprehensive guide will break down each operation, offering clear explanations, examples, and helpful strategies to ensure you develop a solid understanding.

Understanding Integers

Before diving into the operations, let's solidify our understanding of integers. Integers are represented on a number line, stretching infinitely in both positive and negative directions. Zero is the central point, with positive integers to the right and negative integers to the left.

• Positive Integers: These are whole numbers greater than zero (e.g., 1, 2, 3, 100, 1000).
• Negative Integers: These are whole numbers less than zero (e.g., -1, -2, -3, -10, -1000).
• Zero: Zero is an integer that is neither positive nor negative.

Visualizing integers on a number line helps in understanding their relative values and the implications during arithmetic operations.

Adding Integers

Adding integers involves combining their values. The key to success lies in understanding the concept of "signed numbers".

Adding Integers with the Same Sign

When adding integers with the same sign (both positive or both negative), add their absolute values (ignoring the signs) and keep the common sign.

Example 1: 5 + 3 = 8 (Both positive, so add and keep the positive sign)

Example 2: -5 + (-3) = -8 (Both negative, so add and keep the negative sign)

Adding Integers with Different Signs

When adding integers with different signs (one positive and one negative), subtract the smaller absolute value from the larger absolute value. The sign of the result is the same as the sign of the integer with the larger absolute value.

Example 3: 7 + (-3) = 4 (Subtract 3 from 7; since 7 is larger and positive, the result is positive)

Example 4: -7 + 3 = -4 (Subtract 3 from 7; since 7 is larger and negative, the result is negative)

Example 5: -12 + 20 = 8 (Subtract 12 from 20; since 20 is larger and positive, the result is positive)

Example 6: 15 + (-25) = -10 (Subtract 15 from 25; since 25 is larger and negative, the result is negative)

Subtracting Integers

Subtraction of integers can be thought of as adding the opposite. This means changing the subtraction problem into an addition problem by changing the sign of the second integer and then following the rules of addition.

The Rule: a - b = a + (-b)

Example 7: 8 - 5 = 8 + (-5) = 3

Example 8: -6 - 2 = -6 + (-2) = -8

Example 9: 4 - (-3) = 4 + 3 = 7 (Subtracting a negative is the same as adding a positive)

Example 10: -9 - (-4) = -9 + 4 = -5

Multiplying Integers

Multiplying integers involves combining the values multiplicatively. The rules for signs are crucial:

• Positive x Positive = Positive: A positive number multiplied by a positive number always results in a positive number. (e.g., 3 x 4 = 12)

• Negative x Negative = Positive: A negative number multiplied by a negative number always results in a positive number. (e.g., -3 x -4 = 12)

• Positive x Negative = Negative: A positive number multiplied by a negative number always results in a negative number. (e.g., 3 x -4 = -12)

• Negative x Positive = Negative: A negative number multiplied by a positive number always results in a negative number. (e.g., -3 x 4 = -12)

Example 11: 6 x 5 = 30

Example 12: -6 x -5 = 30

Example 13: 6 x -5 = -30

Example 14: -6 x 5 = -30

Dividing Integers

Division of integers follows similar rules regarding signs as multiplication:

• Positive ÷ Positive = Positive: (e.g., 12 ÷ 3 = 4)

• Negative ÷ Negative = Positive: (e.g., -12 ÷ -3 = 4)

• Positive ÷ Negative = Negative: (e.g., 12 ÷ -3 = -4)

• Negative ÷ Positive = Negative: (e.g., -12 ÷ 3 = -4)

Example 15: 20 ÷ 4 = 5

Example 16: -20 ÷ -4 = 5

Example 17: 20 ÷ -4 = -5

Example 18: -20 ÷ 4 = -5

Remember that division by zero is undefined.

Combining Operations: Order of Operations (PEMDAS/BODMAS)

When faced with expressions involving multiple operations, remember the order of operations:

PEMDAS/BODMAS:

• Parentheses/ Brackets
• Exponents/ Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Example 19: 3 + 4 x 2 - 6 ÷ 2 = ?

1. Multiplication and Division (left to right): 4 x 2 = 8; 6 ÷ 2 = 3
2. The expression becomes: 3 + 8 - 3
3. Addition and Subtraction (left to right): 3 + 8 = 11; 11 - 3 = 8
4. Answer: 8

Example 20: (5 - 2) x 4 + 6 ÷ 2 = ?

1. Parentheses: 5 - 2 = 3
2. The expression becomes: 3 x 4 + 6 ÷ 2
3. Multiplication and Division (left to right): 3 x 4 = 12; 6 ÷ 2 = 3
4. The expression becomes: 12 + 3
5. Addition: 12 + 3 = 15
6. Answer: 15

Practice Problems

The best way to master integer operations is through consistent practice. Try these problems:

1. -15 + 7 = ?
2. 12 - (-8) = ?
3. -9 x -6 = ?
4. 36 ÷ (-4) = ?
5. -2 + 8 x 3 - 10 ÷ 5 = ?
6. (4 + (-7)) x ( -2 - 3) = ?
7. -100 + 25 - (-50) x 2 ÷ 10 =?

Conclusion

Adding, subtracting, multiplying, and dividing integers are fundamental arithmetic skills. By understanding the rules governing signs and the order of operations, you can confidently tackle a wide range of mathematical problems. Consistent practice and a solid grasp of these foundational concepts will significantly enhance your mathematical abilities and prepare you for more advanced topics. Remember to utilize the number line as a visual aid to solidify your understanding. Continue practicing and refining your skills to build a strong foundation in mathematics!