Introduction to Mathematics

Paolo Lorenzo BautistaSpecial thanks: Pauline Mangulabnan

De La Salle University

Short Exercise. TRUE or FALSE1. The symbols + and – were introduced before

the 1200s.2. It is possible to count without numbers. 3. The multiplication symbol was introduced by

William Oughtred in the 1400s. 4. There were already evidence of counting

during the time of Christ.5. Famous mathematicians believed that

mathematics is never spiritual.6. Animals can count.

Short Exercise. TRUE or FALSE1. The symbols + and – were introduced before

the 1200s.2. It is possible to count without numbers. 3. The multiplication symbol was introduced by

William Oughtred in the 1400s. 4. There were already evidence of counting

during the time of Christ.5. Famous mathematicians believed that

mathematics is never spiritual.6. Animals can count.

False

False

False

True

True

True

• Among the oldest direct evidence of human counting is a baboon’s thigh bone marked with 29 notches. (150,000 years ago)

• They used different sort of things to count like pebbles, scratches, tallies, straight segments, curves, etc.

There were already evidence of counting during the time of Christ.

Even angles were used!

• Famous Hindu Numerals

• The symbols for addition and subtraction first appeared in 1465 in Robert Muller’s paper.

• However, the downward stroke of addition is not so vertical.

The symbols + and – were introduced before the 1200s.

• In 1631, the multiplication symbol (x) was introduced by William Oughtred in his book, Keys to Mathematics.

• He invented the slide rule (early form of a calculator)

The multiplication symbol was introduced by William Oughtred in the 1400s.

• Srinivasa Ramanujan (INDIAN)– An equation means nothing to me

unless it expresses a thought of God

• Carl Friedrich Gauss– I proved a theorem not by dint of

painful effort but by the grace of God

• George Cantor, Blaise Pascal, John Littlewood, etc.

Famous mathematicians believed that mathematics is never spiritual.

• Counting facilities are found in animals like squirrels, rats, chimpanzees, and other pollinating insects.

• They can distinguish numbers through visual patterns and can produce signals, react to rewards and match dots with numbers to show such recognition.

Animals can count.

H. Kalmus (1964)Tetsuro Matsuzawa (Kyoto)Michael Beran (2003)

HOW CAN YOU COUNT WITHOUT NUMBERS?

Without counting or simply using your fingers, represent quantities from zero to fifty using your body. How are you

going to represent each quantity? Your system must be possible even for counting until a thousand.

How can you represent the time of the day without using numbers? How will you show 3:00 pm? 7:00 pm? 2:00 am? It does not have to be exact to the last minute. What materials

will you be using?

Reflection:

• Why do you think people study mathematics?

• Why do we need people to study mathematics?

Mathematics

“The subject in which we never know what we are

talking about, nor whether what we are saying is

true.”

Bertrand Russell

“Mathematics is the classification and study of all

possible patterns.”

Walter Warwick Sawyer

“The abstract science which investigates

deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and

which includes as its main divisions geometry,

arithmetic, and algebra.”

Oxford English Dictionary

WHAT HAS MATH BROUGHT TO THE WORLD TODAY?

The hypercube: fictional? What is it? Does it exist?

Logic and the mathematical imagination has given a product of the mind a (palpable) reality of its own!

‘True facts about imaginary things’

5-dimensional hypercube

9-dimensional hypercube

What is a 248-dimensional object?

The Lie group E8

In 2006, a physicist proposed that the 248-dimensional E8 is the fabric of the universe.

Mathematicians enjoy the activity of the mind which leads to the most breathtaking realities of today.

From a beautiful idea, math transcends our understanding of the universe.

“imagination is more important than knowledge”

Math is found everywhere! It runs many aspects of our living.

Cars run on engines thatrun on the principles ofcalculus.

CD players, audio and video players, digital cameras and your cell phones use coding theory, linear algebra and geometry.

Internet routing protocols and search engines

use graph theory and linear algebra to manage and efficiently access information on the web.

GPS: Algebra, geometry, trigonometry, statistics and linear algebra

Fractals and Partial Differential Equation

Animated characters are usually constructed from a grid of points with a means to interpolate between the grid points and produce a natural, smooth appearance.

HARMONIC FUNCTIONS

Mathematicians like Stanley Osher, Yves Meyer, Daniel Spielman and Alberto Adrego Pinto are examining areas of life where maths can find applications, including modelling of human behaviour.

Number theory, coding theory and cryptography