The Nature of Mathematics - Video Notes

A set of fill-in-the-blank flashcards covering key concepts from the lecture notes on mathematics, sequences, patterns, and the Fibonacci-Golden Ratio.

Mathematics is the study of , structure, space, and change.

quantity

Mathematics involves reasoning and quantitative calculation.

logical

Mathematics deals with numbers, shapes, and relationships between them, using abstract concepts and deduction.

logical

Mathematics is invented to link the meaning of pattern which results in experiences associated with counting, sequences, and .

regularities

Mathematics has its own symbols, syntax, and rules characterized as precise, concise, and powerful .

language

The four basic concepts emphasized in mathematics are sets, , relations, and binary operations.

functions

It includes elementary logic, connectives, quantifiers, negation and with formality.

variables

Number patterns are lists of numbers that follow a specific or pattern.

rule

Arithmetic sequences have a constant between consecutive terms.

difference

Geometric sequences have a constant between consecutive terms.

ratio

Other patterns include those based on squares, cubes, Fibonacci numbers, or numbers.

prime

The nth term of an arithmetic sequence is given by an = a1 + (n-1) .

d

In an arithmetic sequence, a1 is the _ term.

first

In an arithmetic sequence, d stands for the difference.

common

In the sequence 1, 5, 9, 13, …, the 13th term is .

49

If the 35th term is 687 and the common difference is 14, the first term is .

211

Sum of the arithmetic series 3 + 7 + 11 + … (up to 25 terms) is .

1275

The next term in 1, 4, 7, 10, 13, 16, 19, 22, 25, … is .

28

The 10th term of the sequence 12, 16, 20, 24, 28, … is .

48

The 7th term of the sequence 6, 2, −2, −6, … is .

−18

The 20th term of the sequence 16, 36, 56, 76, 96, … is .

396

The 5th term of the sequence 72, 54, 36, 18, … is .

0

The 9th term of the sequence 33, 76, 119, 162, … is .

377

The 9th term of the sequence 21, 77, 133, 189, … is .

469

The nth term of a geometric sequence is given by an = a1 * r^(n-1), where r is the ratio.

common

The 10th term of the geometric sequence 1, 3, 9, 27, … is .

19683

The 15th term of the geometric sequence 1, 1/2, 1/4, 1/8, … rounded to 5 decimals is .

0.00006

The 13th term of the infinite geometric sequence 1, 1/4, 1/16, 1/64, … is approximately (to 5 decimals).

0.00000

A square number is the square of an .

integer

A cube number is the cube of an .

integer

In Pascal's Triangle, each number is the of the two numbers above it.

sum

A spiral winds around a center and moves closer to or farther from the .

center

The first eleven numbers in the Fibonacci Sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, __.

89

The Golden Ratio is denoted by the Greek letter and is approximately equal to .

phi; 1.618

Solving the ratio equation (x+1)/x = x/1 leads to the quadratic equation .

x^2 − x − 1 = 0

The Golden Ratio provides the most aesthetically pleasing proportion of sides of a .

rectangle

Leonardo da Vinci contributed to the concept of proportions.

divine

The Fibonacci numbers appear in nature as seen in the petals of flowers; for example, a calla lily has 1 petal, trillium 3 petals, hibiscus 5 petals, cosmos 8 petals, corn marigold 13 petals, etc. These are the first eleven numbers in the Sequence.

Fibonacci

The Golden Ratio is approximately equal to .

1.618

In the Renaissance, the golden ratio was associated with aesthetically pleasing proportions in art and architecture, such as De divina proportione by and Leonardo’s work.

Luca Pacioli

Prime numbers in order start as: 2, 3, 5, 7, 11, 13, __.

17

The term for a tiling of a plane using tiles with no overlaps or gaps is called a .

Tessellation

A spiral is a curve that winds around a center and can move closer to or farther from the .

center

Translational symmetry means the pattern remains unchanged after being shifted by a certain distance in a specific .

direction

In a sequence, regularity is defined as when the same thing happens in the same .

circumstances

Symmetry means an object is invariant under certain transformations such as reflection, rotation, or .

scaling

Bilateral symmetry is also called symmetry.

mirror

Radial symmetry features parts arranged around a central .

axis

The equation for the Golden Ratio arises from letting the shorter segment be 1 and the longer segment be x, leading to (x+1)/x = x/1; this yields the quadratic .

x^2 − x − 1 = 0