MATHEMATICAL SYMBOLS

The symbols are the foundation of every culture. They are among the greatest signs of human civilisation, carrying through the centuries humanity's intelligence, knowledge and observations of life. Through symbols, people express ideologies, beliefs and social structures. They are powerful metaphors for humanity's ability to create an abstract metalanguage that serves as a bridge between knowledge and people. A symbol may be a mark, sign, object, action, sound, colour or word that represents ideas, beliefs or relationships between them. Symbols allow people to go beyond what is immediately visible by creating connections between otherwise different concepts and experiences. Symbolism occurs when something represents an abstract idea or concept. Some symbols arise from personal experience, while others are inherited through culture. Language is the most widely used symbolic system. For example, the letters of an alphabet symbolise the sounds of a spoken language. Language is an important source of continuity and cultural identity. All communication is achieved through symbols. The world is filled with them. A red octagon symbolises "STOP". On a map, a blue line may represent a river. Numerals are symbols for numbers. Alphabetic letters represent sounds. Personal names symbolise individuals. A red rose symbolises love and compassion. The variable x in a mathematical equation may represent the position of a particle in space. Sports uniforms, company logos and traffic signs are all symbols. In many cultures, a gold ring symbolises marriage. Some symbols are highly practical; stop signs, for example, provide clear instructions. Gestures also carry symbolic meaning. In cartography, symbols form the legend of a map. The academic study of symbols and signs is called semiotics. Semiotics is closely connected with both linguistics and psychology. It studies not only what a symbol means, but also how it acquired its meaning and how it functions within society. Symbols are tools of complex communication and often have several layers of meaning. Language is a symbolic system through which people communicate and preserve culture. Mathematics is also a symbolic system. How do we know that one symbol belongs to mathematics rather than music or the French language? The question may sound strange because our general cultural knowledge allows us to recognise immediately which field a symbol belongs to. Every branch of human knowledge has its own expressive language and its own symbolic system. Suppose you want to become an artist. Talent is certainly important, but before discovering whether you possess artistic talent, you first have to learn how to draw with a pencil. As your confidence grows, you begin using colours. In other words, you first learn the expressive language of symbols through which artists communicate. Only then can you use this language to express your own emotions, ideas and attitudes towards yourself and the world around you. The same applies to music. You may sing beautifully without knowing musical notation, and your family may be very proud of your voice. However, if you wish to play an instrument professionally or compose music, you must first learn the symbolic language of musical notes through which musicians record and communicate music. Learning French follows exactly the same principle. Before you can read and write French, you must first learn its alphabet, understand how letters combine into words, master grammar and acquire vocabulary. Without knowing the letters, reading is impossible. Mathematics is no different. It also has its own language and symbolic system through which mathematical ideas are expressed and recognised. The language of mathematics consists largely of symbols. Without these symbols, mathematical operations would be almost impossible. Sometimes the smallest things prove to be the most important. Representing relationships between numbers symbolically not only simplifies calculations but also makes mathematical ideas much easier to understand than lengthy verbal explanations. In simple arithmetic, the plus sign, the minus sign and the multiplication sign are all symbols. We need them to express mathematical operations clearly. It would be very inefficient always to write "one plus one equals two" instead of simply writing 1 + 1 = 2. Mathematics is built mainly upon two foundations: numbers and symbols. We encounter symbols in arithmetic, algebra, geometry, probability, statistics and every other branch of mathematics. Decimals and fractions, for example, are symbols representing parts of a whole. Without symbols, mathematics simply could not exist because so much of it deals with abstract ideas. How could we solve for x in algebra without using the symbol x? It would be impossible. Mathematical symbols are indispensable tools for communicating mathematical knowledge effectively. Without them, every calculation would require long verbal descriptions explaining each step. Mathematical symbols are also universal. Almost every country uses the same mathematical notation, making mathematics an international language. We may divide mathematical symbols into two groups. The first group contains the so-called technical symbols. They represent mathematical operations and relationships that could otherwise be expressed using words, phrases or even complete sentences. Most of these symbols are familiar from school mathematics, and even people who are not mathematicians use many of them in everyday life. Professional mathematicians use many additional symbols because they work with more advanced and abstract concepts. Computer scientists, physicists, engineers, logicians and many other specialists also rely on these symbols. Since computers have become an essential part of everyday life, basic computer skills also require familiarity with many mathematical symbols. In other words, whenever we use computers, we are constantly using part of the language of mathematics. The second group consists of numbers themselves. This group deserves special attention because numbers are among the greatest cultural and scientific achievements in human history. It took humanity thousands of years to invent symbols that represent increasing quantities in the world around us. Today this seems simple and obvious, but reaching agreement on these symbols required centuries of development across many different cultures. I am sure that we all possess this mathematical and cultural knowledge about numerical symbols, and we use them every day in both our professional and personal lives. Let us therefore begin with the first group of mathematical symbols. These symbols exist by international convention and long-standing tradition. Throughout history, mathematicians from many countries gradually agreed to use the same symbols with the same meanings. Here are some common mathematical symbols: + — Addition. Example: 3 + 7 = 10 − — Subtraction. It is also used to indicate negative numbers. Example: 5 − 2 = 3 × — Multiplication. Example: 4 × 3 = 12 ÷ or / — Division. Example: 20 ÷ 5 = 4 ( ) — Brackets used for grouping mathematical operations. Example: 2(a − 3) [ ] — Square brackets used for grouping more complex expressions. Example: 2[a − 3(b + c)] ∞ — Infinity. It represents the idea that numbers continue without end because another number can always be added. = — Equals. Example: 1 + 1 = 2 ≈ — Approximately equal to. Example: π ≈ 3.14 ≠ — Not equal to. Example: π ≠ 2 <, ≤ — Less than; less than or equal to. Example: 2 < 3 >, ≥ — Greater than; greater than or equal to. Example: 5 > 1 √ — Square root. Example: √4 = 2 ° — Degree. Used for angles and temperature. Example: 20° ⇒ — If... then... Example: If A and B are odd numbers, then A + B is an even number. ⇔ — If and only if; equivalent to. Example: x = y + 1 ⇔ y = x − 1 % — Percent. It represents a fraction out of one hundred. Example: 50% = 50/100 Let us not forget that today we have the privilege of using all these symbols freely. Generations of mathematicians and logicians worked hard to develop and standardise them. Many mathematical symbols arose naturally from the need to express logical relationships clearly and efficiently. We live in a colourful symbolic world. Every day we are surrounded by countless signs and symbols from different areas of life. Gradually we learn to understand their language and to communicate ideas through them. Some symbols speak to us particularly deeply. The red heart, for example, is universally recognised as a symbol of love and affection because people have long associated the human heart with emotions. We also use many spiritual symbols, depending on our religious beliefs and cultural traditions. There is a special relationship between symbols and human beings. Symbols are created by people, yet they profoundly influence people's lives. The relationship works in both directions. Over centuries, humanity has invested its hopes, beliefs and values in symbols. As a result, symbols become filled with collective meaning and emotional power. Many of the most important spiritual symbols form part of our cultural heritage and inherited wisdom. Often they become objects of prayer and devotion. Because of this, they acquire a special symbolic power that inspires hope, faith and spiritual growth. The human ability to invent symbols and communicate through them is one of the strongest proofs of our capacity for abstract thought. In many ways, it is one of the qualities that brings us closer to the divine. That is where we truly belong. Activities Activity 1: Can you think of other situations in which brackets are used? Activity 2: Practise using the percent sign (%). Did you use percentages today while shopping? Activity 3: If you had to invent a new symbol for an action, idea or process, what would it look like? What would it represent? Activity 4: Why do you think a ring has become the symbol of marriage? What does it represent? Activity 5: Can you make a list of some spiritual symbols from your own culture? (Elena S. Lyubenova)