(Use this for context) The diameter of Earth is approximately 1.27 x 10⁷ meters.

Question

The tallest building in the world, the Burj Khalifa in Dubai, has a height of about 8.28 x 10² meters.

(Now this is the question)
Explain whether the number 1.27 x 10⁷ is rational or irrational, and justify your reasoning.
(write this as a 8th grader mid-way through the school year.)

Answer 1

To determine if the number \( 1.27 \times 10^7 \) is rational or irrational, we first need to understand what those terms mean.

A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers (whole numbers), and \(b\) is not zero. For example, \( \frac{1}{2} \), \( \frac{3}{4} \), and even whole numbers like \(5\) (which can be written as \(\frac{5}{1}\)) are rational numbers.

An irrational number, on the other hand, cannot be expressed as a simple fraction. These numbers have decimal representations that go on forever without repeating. Examples include numbers like \(\pi\) (which is about 3.14159...) and the square root of 2.

Now, let’s analyze the number \( 1.27 \times 10^7 \):

1. Break it down: The number \( 1.27 \) is a decimal, and \( 10^7 \) means that we are multiplying \( 1.27 \) by \( 10,000,000 \) (which is the same as moving the decimal point 7 places to the right). This gives us \( 12,700,000 \).

2. Check if it can be a fraction: Since \( 1.27 \) can be written as \(\frac{127}{100}\), we can express \( 1.27 \times 10^7 \) as: \[ 1.27 \times 10^7 = \frac{127}{100} \times 10^7 = \frac{127 \times 10^7}{100} = \frac{127 \times 10^5}{1} = 12,700,000 \] Here, \( 12,700,000 \) is a whole number.

3. Conclusion: Since we can express \( 12,700,000 \) (and also \( 1.27 \)) as a fraction of two integers, it means that \( 1.27 \times 10^7 \) is a rational number.

In summary, \( 1.27 \times 10^7 \) is rational because it can be expressed as a fraction made up of integers. It equals \( 12,700,000\), which confirms that it is a rational number since it can clearly be written as \(\frac{127000000}{1}\).

Answer 2

write it as a 8th grader please!

Answer 3

To figure out if the number \( 1.27 \times 10^7 \) is rational or irrational, we need to understand what those terms mean.

A rational number is a number that can be written as a fraction like \(\frac{a}{b}\), where \(a\) and \(b\) are whole numbers (called integers), and \(b\) isn’t zero. For example, numbers like \( \frac{3}{4} \) and \( 5 \) (which can be written as \(\frac{5}{1}\)) are rational.

An irrational number, on the other hand, cannot be written as a simple fraction. These are numbers that have decimal points which go on forever without repeating, like \(\pi\) (which is about 3.14...) or the square root of 2.

Now, let’s look at \( 1.27 \times 10^7 \):

1. Understanding the number: The number \( 10^7 \) means \(10\) multiplied by itself 7 times, which is \(10,000,000\). So when we multiply \( 1.27 \) by \( 10^7\), we’re actually moving the decimal point in \( 1.27 \) seven places to the right. This gives us \( 12,700,000\).

2. Can it be a fraction?: We can also break down \( 1.27 \). The decimal \( 1.27 \) can be written as \(\frac{127}{100}\). So when we multiply it out: \[ 1.27 \times 10^7 = \frac{127}{100} \times 10^7 = \frac{127 \times 10^7}{100} = 12,700,000 \] Since \( 12,700,000\) is a whole number, we can write it as a fraction too, like \(\frac{12,700,000}{1}\).

3. Making a decision: Since we can write \( 1.27 \times 10^7\) as a fraction of two integers, that means it is a rational number.

In conclusion, \( 1.27 \times 10^7 \) is a rational number because we can express it as a fraction and it equals \( 12,700,000\), which is a whole number. So, we can definitely say it is rational!