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Chapter 1: Problem 23

Use the properties of division to simplify the expression, if possible. \(0 \longdiv { 9 }\)

Short Answer

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Step by step solution

01

Identify the Division Expression

The given expression is the division of 0 by 9, written as \(0 \div 9\).
02

Understand Division with Zero

Recall that dividing zero by any non-zero number results in zero. This is because any number times zero equals zero.
03

Apply the Rule

Applying the rule, \(0 \div 9 = 0\). Since 9 is a non-zero number, the result of the division is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividing by Zero
Understanding what happens when you divide by zero is essential in mathematics. It's a special case that often confuses students. When you divide any number by zero, the result is undefined. Why is this? Let's imagine you have a number, like 5, and you try to create groups of zero items each. It's impossible because you can't form any groups at all.
This is why mathematicians say that division by zero is not allowed. It's undefined and doesn't fit into our number system. Always remember:
  • Any number divided by zero is undefined.
  • Zero divided by any non-zero number equals zero.
Mathematical Rules
Mathematical rules are consistent principles that help us solve problems. Even when dealing with complex calculations, these rules guide us to the correct answer. Let's go over a few important ones related to division and zero:
  • Zero Property of Multiplication: Any number multiplied by zero is zero. Therefore, if you divide zero by any non-zero number, the result will be zero. This is because reversing the process (multiplying zero by a number) always leads to zero.
  • Division by Non-zero Numbers: When you divide a non-zero number by another non-zero number, the result is just a fraction or another number. However, zero divided by a non-zero number is always zero because zero divided into any number of parts is still zero.
  • Undefined Division: Dividing any number by zero does not produce a result that fits into our number system. Hence, we say it is undefined.
These rules help us keep calculations consistent and predictable.
Simplifying Expressions
Simplifying expressions makes math easier and more understandable. Here's how it works, specifically for division:
  • Identify Components: Break down the expression into its core components. In our case, the expression to simplify is \(0 \div \ 9\).
  • Apply Division Rules: Use the known rules to simplify. Recall that any number divided by a non-zero number equals zero, so \(0 \div \ 9 = 0\).
  • Simplified Result: After applying the rules, write down the simplified result. Here, \(0 \div \ 9\) simplifies directly to zero.
Simplifying makes it easier to see the value and understand the math behind it. Always follow these steps for clear and correct simplifications.

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