Question

Prep Exercise 6 When dividing two num bers that have the same sign, the result is ___. Prep Exercise 7 When dividing two numbers that have different signs, the result is ___. divide. $$\frac{-124}{-4}$$

Short answer

Answer: 31

Step-by-step solution

Prep Exercise 6

When dividing two numbers with the same sign (both positive or both negative), the result is positive.

Prep Exercise 7

When dividing two numbers with different signs (one positive and one negative), the result is negative.

Example: Dividing -124 by -4

Since both numbers are negative, according to Prep Exercise 6, the result will be positive. To calculate the division, we can ignore the signs and divide the absolute values: $$\frac{-124}{-4} = \frac{124}{4}$$ Now, divide 124 by 4: $$\frac{124}{4} = 31$$ The result is positive as per the rule stated in Prep Exercise 6: $$\frac{-124}{-4} = 31$$

Key concepts

Integers

Integers are whole numbers that can be positive, negative, or zero. They're like counting numbers, but they include zero and the negatives. You can imagine them as points on a number line where the middle is zero.

Some examples of integers are:

• -3: A negative integer
• 0: Neither negative nor positive, just zero
• 8: A positive integer

Understanding integers is crucial because they are fundamental in arithmetic operations like addition, subtraction, multiplication, and division. Knowing how to work with integers helps solve everyday problems and mathematical puzzles.

Signs in Division

The signs of the numbers you are dividing play a big role in determining the sign of the result. Here are the simple rules to remember:

• If both numbers you're dividing have the same sign (both positive or both negative), the result is positive.
• If the numbers you're dividing have different signs (one positive and one negative), the result is negative.

This can be thought of as a way to tell if the balance of signs is fair or offset. So, when dividing:

- Positive ÷ Positive = Positive - Negative ÷ Negative = Positive - Positive ÷ Negative = Negative - Negative ÷ Positive = Negative Remembering these rules can make dividing integers much easier.

Absolute Value

Absolute value is essentially the distance a number is from zero on the number line, without considering direction. So, it's always a positive number or zero. When we write the absolute value of a number, we use two vertical lines, like this:

• \(|-5| = 5\)
• \(|7| = 7\)
• \(|0| = 0\)

Absolute value is particularly useful in division because it allows us to handle the numbers straightforwardly without worrying about the signs initially. In the exercise, dividing the absolute values of the numbers \(\frac{124}{4}\) allows us to focus simply on the computation and add the sign afterwards based on the division rules of integers. This simplifies arithmetic considerably when working with different signed integers.