The binomial theorem is a powerful tool in applied biology for approximating roots of numbers and calculating small changes in measurements, which are common in biological data analysis. This chapter explores how binomial expansions help compute roots and errors efficiently, essential for precision in experiments and data interpretation in Kenyan laboratories, farms, and research institutions. Understanding these applications enhances accuracy in measurements such as enzyme concentrations, growth rates, and population genetics.

7.1 Roots of numbers using binomial theorem

Binomial expansions allow us to approximate roots of numbers that are close to 1 by expressing the root as a binomial raised to a fractional power. This method is especially useful when dealing with biological quantities that need quick approximations without complex calculators, such as estimating growth factors or concentrations near a standard value.

7.1.1 Pascal's triangle

Pascal's triangle provides the coefficients for the expansion of a binomial expression raised to any natural number power. Each row corresponds to the coefficients of the expansion terms, which are essential when applying the binomial theorem to approximate roots or powers.

The first few rows of Pascal's triangle are:

\[ \begin{array}{cccccc} & & 1 & & & \\ & 1 & & 1 & & \\ 1 & & 2 & & 1 & \\ 1 & 3 & & 3 & & 1 \\ 1 & 4 & 6 & 4 & 1 & \\ \end{array} \]

The \(n\)-th row gives coefficients for the expansion of \((a + b)^n\).

Worked Examples

Example 1: Find the coefficients for the expansion of \((a + b)^3\).

Given: \(n = 3\)

The coefficients are the 3rd row of Pascal's triangle: 1, 3, 3, 1.

Answer: Coefficients are 1, 3, 3, 1.

Example 2: Write the expanded form of \((a + b)^4\) using Pascal's triangle.

Given: \(n = 4\), coefficients from 4th row: 1, 4, 6, 4, 1

\[ (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \]

Answer: \(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\)

Example 3: Using Pascal's triangle, find the coefficient of the \(b^2\) term in the expansion of \((a + b)^5\).

Given: \(n=5\), row 5 coefficients: 1, 5, 10, 10, 5, 1

The term with \(b^2\) corresponds to \(a^{5-2}b^2 = a^3b^2\), coefficient is 10.

Answer: Coefficient is 10.

7.1.2 Approximating roots with binomial expansion

To approximate \(\sqrt[n]{1 + x}\) where \(|x| \ll 1\), we write:

\[ (1 + x)^{\frac{1}{n}} = 1 + \frac{1}{n}x + \frac{\frac{1}{n} (\frac{1}{n} - 1)}{2!} x^2 + \frac{\frac{1}{n} (\frac{1}{n} - 1) (\frac{1}{n} - 2)}{3!} x^3 + \cdots \]

This series provides a way to estimate roots of numbers close to 1 by truncating after a few terms.

Worked Examples

Example 1: Approximate \(\sqrt{1.04}\) using binomial expansion up to the second term.

Given: \(n=2\), \(x = 0.04\)

\[ (1 + 0.04)^{\frac{1}{2}} = 1 + \frac{1}{2} \times 0.04 = 1 + 0.02 = 1.02 \]

Answer: \(\sqrt{1.04} \approx 1.02\)

Example 2: Approximate \(\sqrt[3]{0.98}\) up to the second term.

Given: \(n=3\), \(x = -0.02\)

\[ (1 - 0.02)^{\frac{1}{3}} = 1 + \frac{1}{3} \times (-0.02) = 1 - 0.0067 = 0.9933 \]

Answer: \(\sqrt[3]{0.98} \approx 0.9933\)

Example 3: Approximate \(\sqrt{1.05}\) including the second-order term.

Given: \(n=2\), \(x=0.05\)

\[ (1 + 0.05)^{\frac{1}{2}} = 1 + \frac{1}{2} \times 0.05 + \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2} \times (0.05)^2 \]

Calculate the coefficients:

\[ \frac{1}{2} = 0.5 \]

\[ \frac{1}{2} - 1 = -0.5 \]

\[ \frac{0.5 \times (-0.5)}{2} = \frac{-0.25}{2} = -0.125 \]

Substitute:

\[ 1 + 0.025 - 0.125 \times 0.0025 = 1 + 0.025 - 0.0003125 = 1.0246875 \]

Answer: \(\sqrt{1.05} \approx 1.0247\)

7.1.3 Applications in biological data

In applied biology, roots approximated by binomial expansions help estimate growth rates, enzyme kinetics, and population genetics parameters when exact calculations are difficult.

Worked Examples

Example 1: A bacterial population increases by 4%. Estimate the square root of the growth factor to find the approximate growth rate per generation.

Given: Growth factor \(1 + x = 1.04\), find \(\sqrt{1.04}\)

Using previous calculation:

\[ \sqrt{1.04} \approx 1.02 \]

Answer: Approximate growth rate per generation is 1.02 (2% increase).

Example 2: The concentration of a nutrient decreases by 3%. Estimate the cube root of the concentration factor to find the approximate change per day over 3 days.

Given: \(1 + x = 0.97\), \(n=3\), \(x = -0.03\)

\[ (1 - 0.03)^{\frac{1}{3}} = 1 + \frac{1}{3} \times (-0.03) = 1 - 0.01 = 0.99 \]

Answer: Approximate daily concentration factor is 0.99 (1% decrease per day).

Example 3: Estimate \(\sqrt[4]{1.08}\) to approximate a four-day compound growth in a crop yield.

Given: \(x=0.08\), \(n=4\)

\[ (1 + 0.08)^{\frac{1}{4}} = 1 + \frac{1}{4} \times 0.08 = 1 + 0.02 = 1.02 \]

Answer: Approximate daily growth factor is 1.02 (2% increase per day).

Chapter Summary

This chapter explored the application of the binomial theorem in various mathematical contexts relevant to science. It began with the use of Pascal's triangle to expand binomial expressions and find roots of numbers by approximating them through binomial expansions. The discussion then moved to the analysis of errors resulting from small changes, introducing absolute error as the difference between measured and true values, relative error as the ratio of absolute error to the true value, and percentage error as the relative error expressed in percentage form. These concepts are essential for quantifying the accuracy of measurements and calculations. Finally, the chapter covered permutation and combination, which are fundamental counting techniques used to determine the number of ways objects can be arranged or selected in specific orders or groups. Together, these topics provide a comprehensive understanding of how binomial expansions and related principles are applied in practical scientific calculations.