How fast is the world population growing?

Exponential functions in mathematics is a term that represents a quantity in which increases without limit based on an exponential function. That function in its simplest form is f(x)=a^x. Exponential functions are functions that grow in a way proportional to their size. The importance of exponential functions in mathematical is plenty; however, I will use them to represent real-world situations in this blog post to understand this concept better. Exponential growth is growth that increases by a constant proportion, and for a period in history, our human population has grown exponentially. An example of exponential functions is the growth in the human population.

Desmos was used to graph the data points to see this relation better. This software would later help graph a function that best fits the data. In Figure 1, when x=0, this represents the year 1951. Whether it’s an exponential function or not, there needs to be a dependent and independent variable when modeling graphs. In this case, the dependent variable will be the value that is growing exponentially, the number of people. While the independent variable is time by year. Looking at the population on earth provided by Worldometer, a website that offers counters and real-time statistics for diverse topics. For the past 70 years, while the population has grown, the growth rate is not proportional to its size as it is more of linear growth. Rather than increasing by a constant change rate, it grows by a similar amount each year. Although, there was a period in history in which our population increased relatively exponentially. If we look at Table 1, it can be seen that the growth rate was reasonably similar during the 25 years between 1951 and 1975.

From the table, through these 25 years, the growth by percentage at which the population was growing ranges between 1.76% and 2.09%. Figure 2 below illustrates these figures in a graph; however, this time, the population is graphed by a hundred million. This will then help determine a function that best fits the data as smaller numbers were used.

By looking at Figure 2, the world population was somewhat growing exponentially. While it does not increase proportionally, there is still a similar trend. From here, an exponential function can be used to model this trend. Population growth follows the general form of:

From Table 1, we know that P_o=2,584,034,261. To make the number more manageable while still indicating small differences, it will be rounded to the hundred million while two decimal places will be included. Therefore, P(0)=25.84. A common growth rate can be found by taking the average values in Table 1; therefore, k = 1.92% = 0.0192 as a percentage can be changed by dividing by 100. This information can be plugged into Formula 1 for a general function.

As mentioned before, e is a numeric value; it is the exponential constant with an approximate value of 2.718. To understand, the constant e is the base of the natural logarithm, with is the base rate of growth shared by all exponentially growing functions. Furthermore, e represents the idea that all continually growing systems are scaled versions of a common rate. When t, which represents the time by year, is inserted into the function, it will determine the population growth at the given year. Assuming the function is correct, to verify, let t=10.

Referring to Table 1, t = 10 represents the year 1961 as the population was 30.91 hundred million. This shows that the function does not accurately predict the value; however, it does come relatively close.

Overall, if we used the function and then found the percentage error between the actual and expected values, the average error would be 0.63%. This is a relatively small error as when t = 10, the 1.29% difference equivalates to around a 40 million people difference. Therefore, a slight difference accounts for a large number of people. To graph the function, we will need to change the function into the general formula for exponential functions. This is because the coefficients used in Formula 1 are not valid.

Formula 3 is similar to Formula 1 as y = P, a = P_o, b = e, x = t because the x value represents that independent variable or x-axis, which in this case is time by year, and k is a coefficient next to x, which shows the rate of growth, producing the following function:

The function graphed in Figure 3 is somewhat a good model to represent the data. It shows that each year the population is growing at a constant rate. However, there are still many noticeable anomalies. The function fits with the values at the beginning and the end of the period but not as much in between. We can determine the accuracy of the function in relation to the expected values by finding the correlation coefficient, which shows r=0.9988, indicating a strong exponential growth correlation which means that the function somewhat accurately models the data. Moreover, the values from the function do predict known real-life values to some extent.

We can use a computer-calculated exponential regression to verify the function further to see how close it came. We can find an exponential function that best fits the data set through technology. The function derived through computer calculating is:

As seen in Figure 4, the two functions are pretty similar; however, upon finding the correlation coefficient, the computer calculated function has a stronger correlation because r=.9996. While both functions indicate a strong exponential growth correlation, positive r values close to 1 relate to a closer relationship between the function and known values. Comparing both functions show that the a values stay relatively similar while the b or e coefficient has changed drastically. This means that the growth factor, a quantity that multiplies itself over time, does not follow the mathematical constant e does not precisely apply to this data set. This makes sense because the greater the base, b value is, the faster the graph rises from left to right. Essentially the function is being vertically stretched. Furthermore, while the computer calculated function presents a slightly more accurate model, both functions sufficiently represent the human population from 1951 to 1975.

As stated before, world population growth is not growing exponentially in recent years as in Figure 5; when Function 2 is graphed onto the population up to 2020, the function deviates from actual values.

When x = 25, which refers to 1976, the function diverges from the known values. Therefore, this function can not accurately represent this whole data set. After 1975, the growth starts to slow down and look more like linear growth. Function 1 does produce reasonable values. However, this is only true for the population between 1951 to 1975. While the function is not entirely accurate, the values received from the function are relatively similar to the known values. One limitation to the model is that the human population is not growing exponentially in recent years. Meaning in Function 2, the function’s domain is limited to the resiriction. Also, there is no reliable data on the human population before 1951. Therefore, the function cannot validly predict the human population, which means that Function 1 can only determine the world population between 1951 and 1975. Furthermore, while the human population increased exponentially at specific periods, Figure 5 shows that the world population does not grow exponentially. Although future trends may differ, exponential growth clearly doesn’t describe the global reality of the twenty-first century as nowadays, linear regression would more accurately model the values,

To conclude, exponential growth is when a value increases by a fixed percentage as the quantity increases, so does that rate at which it grows. Many factors influence population growth or plateau. This mathematical concept is fundamental as it can help predict real-life such as population growth. Medical and technological advances have helped increase the population while global pandemics have limited growth. Either way, in theory, population growth should increase exponentially; through this investigation, we can see that this has only been true for specific periods in history. Moreover, between 1951 to 1975, the population was growing at could be modeled through the presented exponential function.