Question

Draw \(10^{1}\) squares on a piece of paper and time yourself as you do it. How long would it take you to draw \(10^{2}\) squares? \(10^{4}\) squares? Estimate how long it would take you to draw \(10^{15}\) squares ( 60 seconds \(=1\) minute, \(5.3 \times\) \(10^{5}\) minutes \(=1\) year). Comment on expressing numbers in an exponential fashion.

Short answer

To estimate the time for 100 (10^2) squares, multiply the time for 10 squares by 10. Multiply the result by 100 for 10,000 (10^4) squares. Extrapolate to find time for 10^15 squares, convert to minutes and then to years, acknowledging that drawing 10^15 squares is unrealistic. Exponential representation simplifies the process of scaling numbers significantly.

Step-by-step solution

Drawing 10 Squares and Timing

First, draw 10 squares on a piece of paper as quickly and accurately as you can and time the duration it takes to complete this task. Make sure to use a stopwatch or timer to accurately record the time.

Calculating Time for 100 Squares

Since it takes a certain amount of time to draw 10 squares, to estimate the time it would take to draw 100 squares, you would multiply that time by 10. This is because 100 squares is 10 times the amount of 10 squares.

Calculating Time for 10,000 Squares

Likewise, to find out the time for drawing 10,000 squares, which is 10 squared or 100 times the amount of 100 squares, you would multiply the time it took to draw 100 squares by 100.

Estimating Time for Drawing 10^15 Squares

To estimate the time for drawing 10^15 squares, use the time it took for the smaller quantities and extrapolate using multiplication. Once you have this time in seconds, convert it to minutes by dividing by 60, to years by dividing by 5.3 x 10^5 minutes, and then discuss the feasibility of your answer.

Commenting on Exponential Representation

Comment on expressing numbers in an exponential format by explaining that it allows for easier representation and understanding of very large quantities. Notice the pattern of multiplication required to go from one power of 10 to the next and emphasize how quickly the number of squares increases despite the small increase in the exponent.

Key concepts

Scientific Notation

Scientific notation is a convenient way to express large or small numbers, which can be particularly useful in disciplines such as chemistry where extreme values are common. It involves writing numbers as the product of two parts: a coefficient and a power of 10. The coefficient is a number greater than or equal to 1 and less than 10, and the power of 10 represents how many times the coefficient must be multiplied or divided by 10. For example, the number 300 can be written in scientific notation as \(3 \times 10^{2}\).

In the context of the exercise, drawing \(10^{1}\) squares is easy to understand and do, but as the exponent increases, the scientific notation helps us to quickly grasp how significantly the quantity increases. Without this notation, it would be cumbersome and prone to errors to write out all those zeroes for, say, \(10^{15}\) squares. Additionally, scientific notation standardizes measurements, which allows scientists and students to communicate findings and calculations more effectively and with less chance of misunderstanding.

Exponential Growth

Exponential growth describes a process where the amount increases at a rate proportional to its current value. In our exercise, the increase in the number of squares follows an exponential pattern: each step involves drawing ten times as many squares as the previous step. This reflects the very nature of exponential growth – rapid increase.

In practical terms, this means that if it takes you 1 minute to draw \(10^{1}\) squares, drawing \(10^{2}\) squares could take you around 10 minutes, assuming a consistent pace. However, as soon as you step up to \(10^{4}\) squares, you aren't just drawing ten times more; you're now up to drawing 100 times more than the initial 10 squares. And by the time you consider \(10^{15}\) squares, you're working with numbers so large that the drawing activity transcends human time scales entirely – illustrating just how quickly exponential growth can escape the bounds of practicality.

Orders of Magnitude

Orders of magnitude provide us with a way to understand the relative size of numbers. Each order of magnitude corresponds to a tenfold increase or decrease in a number. In simple terms, going from \(10^{1}\) to \(10^{2}\) is a jump of one order of magnitude. Going from \(10^{1}\) to \(10^{3}\) is a jump of two orders of magnitude, and so on. This system helps us to compare numbers and comprehend the scale of changes or differences more intuitively.

In the exercise, each step up in the exponent (the number above the 10 in scientific notation) represents an increase by one order of magnitude. It’s a straightforward way to see how much larger \(10^{4}\) is compared to \(10^{2}\) – it's two orders of magnitude greater. Learning about orders of magnitude is critical for students as it aids in visualizing the exponential growth in a tangible way and puts into perspective the significance of each incremental change in the exponent.