Mathematical Notation, Symbols and Operators

In order to save ourselves time, we often use symbols as shorthand. We look at three new symbols Δ, Σ and Π, which are used to describe operations, just like (+), (−), (×), and (÷).

The Delta Δ Operator

We use the Greek capital letter Δ (called delta) to represent the difference (or change) between the start and end values of a quantity. For any quantity \(A\):

Note

If \(\Delta A\) is positive, then there is an increase in \(A\) and if \(\Delta A\) is negative, then there is a decrease in \(A\).

\[ \Delta A = A_{\textrm{final}} - A_{\textrm{initial}} \]

where \(A_{\textrm{final}}\) is the final amount of \(A\) and \(A_{\textrm{initial}}\) is the initial amount of \(A\), so \(\Delta A\) is equal to the difference between these to values.

Change in Absorbance

If the optical abosrbance \(Abs\) increase from 0.65 to 1.35 during a reaction, what is \(\Delta(Abs)\)?

Solution: First recall that:

\[ \Delta(Abs) = Abs_{\textrm{final}} - Abs_{\textrm{initial}} \]

From the question, \(Abs_{\textrm{initial}} = 0.65\) and \(Abs_{\textrm{final}} = 1.35\). Hence \(\Delta(Abs) = 1.35 - 0.65 = 0.70\).

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This is straightforward in Python

abs_initial = 0.65
abs_final = 1.35

delta_abs = abs_final - abs_initial
delta_abs

0.7000000000000001

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The Sigma Σ Operator

The Greek capital letter Σ (called sigma) is used to represent a sum, normally long sums which would take a while to write out completely. The general form for a sum using sigma is:

\[ \sum^{n}_{i=1}{X_i} = X_1 + X_2 + X_3 + \ldots + X_n \]

Below, we have which each of the terms in the sum means:

• \(X_i\) are the numbers that are being summed; it could be an expression involving \(i\).

• \(i\) is called the summation index; it identifies each term in the sum.

• \(i=1\) tells us to start summing at \(i=1\).

• \(n\) tells us to stop summing when we have reached \(i=n\).

For a numerical explanation, consider take the sum \(1 + 2 + 3 + 4 + 5\). This can be simplified into the sum notation as \(\sum_{i=1}^{5}i\). From the notation, we can see that we are summing the number \(i\) from when \(i\) is equal to 1 to 5.

Note

The superscript 2 in this example is the power. We will visit that later.

Example

What is the value of \(\sum_{i=4}^7{i^2}\)

Solution: To begin, we ‘read’ what the operation is telling us to do. We know it is a sum from the Σ and the terms we are summing are the squares of the index. We are starting with 4 and ending at 7 (and assum that each number is an integer). This then will give us:

\[ \sum_{i=4}^7{i^2} = 4^2 + 5^2 + 6^2 + 7^2 = 16 + 25 + 36 + 49 = 126 \]

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In Python, there are two ways to perform a summation, the first (and significantly slower) approach is to use a loop, for example.

total = 0
for i in range(4, 8):
    total += i ** 2
print(total)

126

The second is to use the numpy library.

Python Note

numpy is a Python library, a library is something that extends the functionality of the Python programming language. Specifically, numpy is excellent for numerical computation. To use these libraries, it is neccessary to import them.

import numpy as np

i = np.arange(4, 8)
total = np.sum(np.square(i))
print(total)

126

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Relative Molecular Mass

The summation index might not be numbers, for example, when finding the RMM (relative molecular mass) of a molecule, we sum the masses of all its constituent elementes. This can be expressed as:

\[ \sum_{i = \textrm{Element}} N_iM_i \]

where \(i\) goes through all the elements present in the molecule. \(N_i\) is the number of element \(i\) atoms in the molecule and \(M_i\) is the RAM (relative atomic mass) of element \(i\).

For example, finding the RMM of methane, CH₄ would look like this:

\[ \sum_{i=\textrm{Elements}} N_iM_i = N_{\textrm{Carbon}}M_{\textrm{Carbon}} + N_{\textrm{Hydrogen}}M_{\textrm{Hydrogen}} = (1 \times 12) + (4 \times 1) = 16 \]

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The above would be implemented with numpy as shown below.

N = np.array([1, 4])
M = np.array([12, 1])

np.sum(N * M)

16

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The Pi Π Operator

Note

An easy way to remember which is which, Sigma = Sum, Pi = Product.

You will be familiar with the Greek letter π (called pi) as the constant 3.1415… But capital pi Π is used as an operator. The operation it represents is similar to the one that the Σ operator does, but instead of adding all the terms, you multiply them. The general form for the product using Π is:

\[ \prod_{i=1}^{n} X_i = X_1 \times X_2 \times X_3 \times \ldots \times X_n \]

All the other notation you should recongise from the Σ operator.

Overall Yield

A chemist is making a species that requires 3 steps. The first step gives a 66 % yield, the second gives a 50 % yield and the third gives a 95 % yield. What is the overall percentage yield for this synthesis?

Solution: We can calculate the overall percentage yield by using the Π operator as the overall yield is the product of all the individual yields.

\[ \textrm{Overall Yield} = \prod_{i=1}^{3}(\textrm{yield of step i}) \]

Plugging in the numbers gives:

\[\begin{split} \begin{aligned} \textrm{Overall Yield} & = \textrm{yield of step 1} \times \textrm{yield of step 2} \times \textrm{yield of step 3} \\ & = 0.66 \times 0.50 \times 0.95 = 0.3135 \end{aligned} \end{split}\]

Hence the overall percentage yield will be 31 % to 2 significant figures.

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In Python, we can use the np.prod function.

import numpy as np

yields = np.array([0.66, 0.5, 0.95])
overall_yield = np.prod(yields)
overall_yield

0.3135

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