Tuples, Indices, and Coordinates

A tuple is an ordered collection.

For example:

(2, 5)

is not the same as:

(5, 2)

The values are the same. The order is different. For a tuple, that matters.

Why Order Matters

The pair $(2, 5)$ can mean:

2 units horizontally
5 units vertically

If we switch the order to $(5, 2)$ , we get a different point.

This is why we need tuples before vectors. A vector is not merely a bag of numbers. The positions of the numbers carry meaning.

MATH-C01-T05-001Exercise: Order check

Enter 1 if $(2, 5)$ and $(5, 2)$ are different tuples.

Your answer

Compute it first, then check your number.

HintCompare positions

Is the first coordinate the same in both tuples?

SolutionDifferent order

Enter 1. In $(2, 5)$ , the first value is 2. In $(5, 2)$ , the first value is 5.

Indices

An index tells us which item we are talking about.

x = (4, 7, 9)

then we might write:

x_1 = 4,\quad x_2 = 7,\quad x_3 = 9

This is one-based mathematical indexing.

The subscript does not multiply. It names position. The symbol $x_2$ means "the second component of $x$ ", not " $x$ times 2."

MATH-C01-T05-002Exercise: Read the index

If $x = (4, 7, 9)$ using mathematical indexing, what is $x_2$ ?

Your answer

Compute it first, then check your number.

HintCount mathematically

In this notation, the first item is $x_1$ .

SolutionSecond component

The second item is 7, so $x_2 = 7$ . In this mathematical notation, the first component is $x_1$ , so $x_2$ means the second component.

Python Indexing

In Python, indexing usually starts at zero:

x = [4, 7, 9]
print(x[0])  # 4
print(x[1])  # 7
print(x[2])  # 9

This is not a contradiction. It is a convention.

Mathematics often writes the first item as $x_1$ . Python writes the first item as x[0].

MATH-C01-T05-003Exercise: Translate Python indexing

In Python, if x = [4, 7, 9], what does x[1] return?

Your answer

Compute it first, then check your number.

HintStart at zero

x[0] is 4, so x[1] is the next item.

SolutionPython position

Python uses zero-based indexing:

x[0] = 4
x[1] = 7
x[2] = 9

So x[1] returns 7. This differs from mathematical indexing because Python starts counting positions at zero.

Coordinates

Coordinates are indexed values with a geometric meaning.

The tuple

(2, 5)

can name a point in the plane:

move 2 units horizontally
move 5 units vertically

The same notation can become a vector in the next chapter.

MATH-C01-T05-004Exercise: Read a coordinate

For the point $(2, 5)$ , what is the vertical coordinate?

Your answer

Compute it first, then check your number.

HintUse the position

In $(x, y)$ , the second coordinate is usually the vertical coordinate.

SolutionSecond coordinate

The point is $(2, 5)$ . The second coordinate is 5, so the vertical coordinate is 5.

Common Trap

Do not mix mathematical indexing and Python indexing without checking.

If a formula writes $x_1$ , it probably means the first mathematical component. If code writes x[1], it probably means the second Python list item.

Next, we compress repeated addition and multiplication with sums and products.