Problem 49 - Algebra for College Students

Question

Everyone has two (biological) parents, four grandparents, eight great- grandparents, 16 great-great-grandparents, and so on. If we put the word "great" in front of the word "grandparents" 35 times, then how many of this type of relative do you have? Is this more or less than the present population of the earth? Give reasons for your answers.

Step-by-Step Solution

Answer

You would have 137,438,953,472 such relatives. This is much more than the present population of the Earth, which is around 8 billion.

Step 1: Identify the Pattern

Notice that each generation doubles the number of ancestors. For example, you have 2 parents, 4 grandparents, 8 great-grandparents, and so on. Mathematically, the number of ancestors in each generation can be represented by powers of 2.

Step 2: Determine the Generation

If we put the word 'great' in front of 'grandparents' 35 times, we are referring to the generation that is 37 generations back (1 generation for parents, 2 for grandparents, and adding 35 more for the 'greats').

Step 3: Calculate the Number of Relatives

To find the number of relatives 37 generations back, use the formula for the number of ancestors: o_{n} = 2^{n}Where o_{n} represents the number of ancestors, and n is the generation number. In this case: n=37.Substitute this into the formula: 2^{37}.

Step 4: Simplify the Calculation

Calculate 2^{37}. Using a calculator: 2^{37} = 137,438,953,472.

Step 5: Compare with Earth’s Population

As of now, the estimated population of the earth is around 8 billion, which is 8,000,000,000.To determine if 2^{37} is more or less, compare: 137,438,953,472 > 8,000,000,000.Thus, 137,438,953,472 is much more than the present population of the earth.

Step 6: Conclusion

The reason why the number of ancestors exceeds the current population of the Earth is due to overlaps and shared ancestors throughout history.

Key Concepts

Generational Doubling

In family lineage, each generation back doubles the number of ancestors. For instance, you have 2 parents, 4 grandparents, 8 great-grandparents, and so forth. This pattern is known as generational doubling. It means that every generation back you go, the number of direct ancestors multiplies by two. The formula to describe this is simple: the number of ancestors in the nth generation, denoted as 2^n. For example, if you go back three generations:

• Parents: 2^1 = 2
• Grandparents: 2^2 = 4
• Great-grandparents: 2^3 = 8

This exponential increase is the cornerstone of understanding how many relatives you have as you trace your ancestry further back in time.

Powers of 2

Powers of 2 play a crucial role in calculating the number of ancestors you have in any generation. A power of 2 means multiplying 2 by itself a certain number of times. For example:

• 2^1 = 2
• 2^2 = 4
• 2^3 = 8
• 2^4 = 16

As you go further back in generations, this exponential growth becomes immense. In the case of 37 generations back, we use the expression 2^37. This calculates the number of ancestors to be 137,438,953,472. This rapid growth rate is what makes the ancestor count so surprising when you go many generations back. By understanding powers of 2, you can more easily see how the number of ancestors quickly expands.

Ancestry Mathematics

Ancestry mathematics involves using math principles to trace and calculate the number of ancestors you have across generations. The key equation here is \(o_{n} = 2^n\). This equation states that the number of ancestors \(o_{n}\) in the nth generation is found by raising 2 to the power of n. For example, if you want to find how many great-great-great-grandparents you have: 1. Count how many generations back that is.2. Use the formula 2^n to calculate. So, if you count 'great' 4 times, you are 6 generations back: parents (1), grandparents (2), great-grandparents (3), etc. So, 2^6 = 64. By mastering this simple formula, you can calculate and understand your ancestry through a mathematical lens.

Comparison with World Population

When comparing the number of your ancestors 37 generations back with the current world population, the numbers are striking. We calculated that 37 generations back, you would have 137,438,953,472 ancestors. Currently, the world's population is about 8 billion (8,000,000,000). By comparing these figures:

• Your ancestors: 137,438,953,472
• World population: 8,000,000,000

It's clear that the number of ancestors far exceeds the present population. This discrepancy exists because many ancestors are shared. Throughout history, individuals tend to have common ancestors, leading to overlaps. So, despite the mathematical formula suggesting a massive number of ancestors, the reality is limited by shared ancestry and overlapping family trees.