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. $$0 \div(-2)$$

Short answer

Also, what is the result of $$0\div(-2)$$? Answer: When dividing two numbers with the same sign, the result will be positive. When dividing two numbers with different signs, the result will be negative. In the case of $$0\div(-2)$$, since 0 is considered positive and -2 is negative, the result is negative. However, as any number divided by 0 equals 0, the final answer is $$0\div(-2)=0$$.

Step-by-step solution

Prep Exercise 6: Dividing Two Numbers with the Same Sign

To divide two numbers with the same sign, we just divide the numbers as usual and the result will be positive.

Prep Exercise 7: Dividing Two Numbers with Different Signs

When dividing two numbers with different signs, divide the numbers as usual, but the result will be negative.

Example Problem: Calculating $$0\div(-2)$$

In order to divide 0 by -2, we can follow these steps: 1. Determine the signs: In this case, the dividend (0) is considered as having a positive sign, and the divisor (-2) has a negative sign. Since they have different signs, the result will be negative. 2. Perform division: Since we're dividing 0 by a non-zero number, the result is 0. Now, combine the result with the sign rule explained in Prep Exercise 7: A negative sign multiplied by a 0 will still be 0. So, $$0\div(-2)=0$$.

Key concepts

Division Rules

When dealing with division, understanding a few basic rules can make the process much simpler. At its core, division is the operation of distributing a number, called the dividend, into equal groups represented by another number, the divisor. The result of this operation is known as the quotient. Here’s a breakdown of some important division rules:

• If both the dividend and divisor are the same numbers, with an exception of 0, their quotient is always 1. For example, dividing 7 by 7 results in 1.
• If the dividend is equal to 0 and the divisor is any non-zero number, the result is always 0. For instance, 0 divided by 5 equals 0 because there’s no quantity to distribute.
• Division by 1 does not affect the dividend. Hence, any number divided by 1 remains unchanged, just like 6 divided by 1 equals 6.

Grasping these rules can provide a solid foundation to tackle more complex problems or specific scenarios like how the sign of the numbers affects the division, which we'll explore next.

Positive and Negative Integers

When it comes to dividing integers, the signs of the numbers involved play a critical role in determining the sign of the quotient. Here's how it works:

• If you divide two integers with the same sign, whether both are positive or both are negative, the quotient is always positive. For example, dividing 8 by 2 gives a positive 4, just as dividing -8 by -2 results in a positive 4 as well.
• On the other hand, if the integers have different signs, the result is negative. So dividing 8 by -2 will give you -4, and dividing -8 by 2 will also result in -4.

Remember, the rule itself is straightforward: like signs result in a positive quotient, and unlike signs result in a negative quotient. This principle applies irrespective of whether you're handling big numbers, small numbers, or zero, which leads us to the special case of division by zero.

Division by Zero

One of the most important rules in mathematics is that division by zero is undefined. This arises because there's no number that you can multiply by 0 to get a non-zero number, which would be necessary to reverse a division. Despite this, let's explore a few scenarios:

• Dividing any non-zero number by zero doesn’t yield a valid number, and thus, it's undefined. Imagine trying to divide a pizza among zero people—it simply doesn’t make sense, mathematically or practically.
• However, dividing zero by any non-zero number is always valid and results in a quotient of 0. For example, as seen in the earlier exercise, dividing 0 by -2 yields 0.

So the takeaway is simple: dividing by zero is not allowed in mathematics, but dividing zero by other numbers is safe and results in zero.