Maths/Reading mathematics

Reading a formula

After thiswhat you will be able to doExpand finite sigma notation into its terms, distinguish bound counters from free indices, and evaluate sums over chosen list positions.

Questionwhat this lesson answersA formula is shorter than the arithmetic it stands for. What do its letters, subscripts, counters and summation sign tell you when you expand it?

Not coveredwhat this lesson leaves outWe stop at finite sums of ordinary numbers. We take the reordering laws of addition as a starting point. We do not decide whether an infinite sum converges.

Five measurements arrive: 4, 7, 3, 6, and 5. You want one total, so you write

Now imagine five hundred measurements. The arithmetic has not changed, but longhand no longer fits on a page. Compact notation stores the same job in less space.

A letter keeps one value

Suppose one measurement has not been chosen. It could later be 4 or 9. Give that unknown number the name ${x}$. The letter simply names whichever number we choose. If ${x = 4}$, every ${x}$ within this piece of work means 4. Changing its value halfway through would change the instructions without warning.

A subscript gives a list position

Return to the list 4, 7, 3, 6, 5. One letter cannot distinguish its five positions. Put a small position number below the letter. That lower marker is called a subscript. We can now name the five values:

Read ${x_3}$ as “x sub three.” It means the value at position 3, which happens to be 3 here. Think of the subscript as a house number on the street named ${x}$.

The lower 3 is not multiplication. It is not a power, where a raised number requests repeated multiplication. It is not part of the letter’s name. The name stays ${x}$, while the subscript selects one position. It does not alter the stored value. Here ${x_2}$ stores 7, not 72 or ${x}$ multiplied by 2.

An index can visit each position

Sometimes the position should change. Let ${i}$ stand for the current position. A letter used for a position is called an index. Then ${x_i}$ means the list value at position ${i}$.

When ${i = 1}$, ${x_i}$ means ${x_1}$, so its value is 4. When ${i = 2}$, it means ${x_2}$, so its value is 7. Within each step, ${i}$ has one value. It is a placeholder filled with each position in turn. The instruction says exactly when its value changes.

The summation sign repeats an addition

Start with the five-number arithmetic again:

There is a shorthand for exactly that:

The large Greek letter ${\sum}$ is called sigma, and it means “add up.” The letter ${i}$ underneath is the counter. The 1 underneath says where that counter starts. The 5 on top says where it stops. The expression ${x_i}$ on the right is what gets added each time. Each number being added is called a term.

Expand the shorthand by replacing ${i}$ with every position from 1 through 5:

The start and stop numbers are called bounds. Starting at 2 and stopping at 4 selects only the middle three positions:

Starting at 4 and stopping at 2 visits no positions. We define such an empty range to have no terms. Adding no terms has total zero:

A bound counter disappears after the sum

Take the full range again. A counter named ${i}$ visits positions 1 through 5. A counter named ${j}$ visits the same positions. Both instructions therefore add 4, 7, 3, 6, and 5.

The sigma controls the counter and supplies every value it needs. A controlled letter is called bound. Its name does not affect the total. This follows from the definition of the shorthand, rather than from a new assumption. When the range is already known, people often shorten this to ${\sum_i x_i = \sum_j x_j}$.

Now add the full total to the value at position 2. The result is ${7 + 25 = 32}$. At position 4, the result is ${6 + 25 = 31}$. One expression can leave that position undecided:

Here ${j}$ is bound by the sigma and visits the whole list. The ${i}$ is not the counter under that sigma. Nothing in this expression chooses its position. A letter whose value still must be supplied is called free. Renaming ${j}$ changes nothing. Changing the free ${i}$ can change the result.

The repeated expression can contain a calculation

Multiply each list value by itself, then add the results:

Multiplying a number by itself is called squaring. A raised 2 tells us to perform that multiplication. The sigma can repeat this whole calculation, not only a bare list entry:

At each position, first square that position’s value. Then add the resulting term to the total.

A sigma sign only tells you to add

The notation does not claim that its terms are related. They could come from measurements or from an arbitrary list. The sign does not guarantee that the final number is useful.

For finitely many ordinary numbers, changing the order does not change the total. That works because ordinary addition lets us swap and regroup terms. It is a property of addition, not a promise supplied by sigma notation.

An unending sum raises another question. Do its running totals settle toward one number? That settling is called convergence. A sigma sign alone does not promise convergence or meaning.