The numbers are one of the greatest achievements of human civilisation and culture. They are a human invention. Numbers are all around us. They are among the few things that almost everyone remembers from school mathematics. We use numerical symbols in every sphere of our lives—not only professionally but also personally. There is hardly anyone who does not know the numbers.
What would life be like without these symbols? If, in some strange way, they suddenly disappeared and people lost all memory of them, humanity would probably follow the same path once again, using lines and combinations of tally marks to represent increasing quantities. People travelled a very long intellectual journey before discovering that the objects around them could be counted and that quantities could increase or decrease. Eventually they also realised that quantities follow a sequence. Later they invented ways to represent situations in which something decreases or is lost. This is how the ideas of positive and negative numbers emerged.
When and where do we use numbers? The answer to this question brings us back to the main purpose of our lessons—the value of mathematics in the everyday lives of people who are not mathematicians.
Let us do a simple calculation. How many times during a single day do we use a number, either as a symbol or as a word? If our profession relies heavily on numbers, then we speak the language of numbers almost all day long. Accountants, economists, tax inspectors, mathematics teachers, computer programmers and many other professionals communicate through numbers. By combining them, they send messages and tell stories that their audience understands perfectly.
People in every other profession also use numbers countless times every day. When we wake up in the morning, we look at the numbers on our alarm clock. Numbers accompany us throughout the day. We see them on our watches, on road signs, on bus timetables and on the prices of products in shops. Whenever someone asks our age, height, weight, year of birth or how many years we have lived somewhere, we answer with numbers.
When we cook, we constantly measure ingredients and express their quantities using numbers. In all these everyday situations, we use numbers either as symbols or as words. Numbers become part of the memories of our lives because so many of our experiences can be expressed through them. Even when we ask ourselves how long it has been since we last saw someone, we immediately begin thinking in numbers.
Numbers are also part of the vocabulary of every natural language. They appear in idioms, proverbs and sayings. If we deliberately tried to count how many times we mention numbers during a single day, even without being mathematicians, we would probably find it impossible because, consciously or unconsciously, numbers are part of the way we perceive the world. Our brains naturally recognise objects as one, two, three or many. The same applies to abstract concepts. Although we cannot see or touch them, we still imagine them as singular or multiple. Everyone has had at least one love in life. Some people have experienced many loves, joys, sorrows, hopes and disappointments. Yet, as far as we know, we each have only one physical life.
The recognition of numbers is closely connected with the idea of counting. Counting means recognising how quantities increase step by step.
How many digits do we use? We work with ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. By combining these ten symbols, we can create infinitely many numbers. We have already introduced the idea of mathematical infinity. No matter how long a number becomes, another digit can always be added. In this lesson we are discussing only whole numbers. Other types of numbers will be introduced later.
What is a whole number? It is usually associated with the idea of something complete and undivided.
Another interesting concept is sequence, something we constantly use in everyday life. Although it is connected with counting, it is mainly used for comparison. We compare things all the time. We want to know which is first, second, third, fourth, fifth, sixth, seventh, eighth, ninth or tenth.
We also constantly arrange numbers according to their natural order. Every day we perform simple operations such as knowing that four comes before five and after three. When we stand in a queue at a shop, for example, we automatically count the people ahead of us to estimate when it will be our turn. In such situations we use our ability to count, compare and arrange—not because numbers themselves are real objects, but because they represent real objects.
When we are shopping, we compare prices and decide whether we have enough money to buy what we need.
Numbers exist to represent something real. But what exactly do they symbolise? There is no single answer. They may symbolise almost everything—or nothing at all.
When we see the symbols 2 or 4, we immediately imagine exactly two or four objects. This precision is one reason why mathematics is considered an exact science. But at the same time mathematics is highly abstract because the same numerical symbol can represent completely different things. For example, "two apples" and "two galaxies" contain exactly the same quantity. Numerically they are equal, but the size and significance of the objects they represent are vastly different. This is one of the remarkable powers of numbers: behind the same symbol can stand almost anything in the visible or invisible world.
What do we imagine when we see the symbol 0? Is it not remarkable that we have a symbol for "nothing"? That is precisely what zero represents—the absence of something. Zero is a brilliant invention because it allows us to express nothingness with a single symbol. We constantly use zero in this sense, from indicating the absence of quantity to expressing abstract ideas such as, "I have zero respect for you."
Yet zero is also magical in another way. It has the extraordinary ability to increase the value of numbers. Write any number on a piece of paper and then add as many zeros after it as you wish. Suddenly the number becomes enormously larger. In this sense, zero seems almost magical, multiplying value with astonishing simplicity.
Numbers are also classified as positive and negative. At first this may sound similar to positive and negative human qualities, but the mathematical meaning is quite different.
What is a positive number? Simply put, it is any number greater than zero. Positive numbers are used whenever we describe quantities that exist.
What is a negative number? It is any number smaller than zero, written with a minus sign. Negative numbers are useful whenever we describe decreases, losses or values below a chosen reference point.
This means that zero is not the smallest number. Although it represents nothing, numbers such as –1, –2 and –10 are even smaller.
For example, compare 5 and –5. The first is positive; the second is negative. Negative numbers are commonly used to represent temperatures below zero on a thermometer. Whether the outside temperature is 20°C or –20°C makes an enormous difference to the clothes we wear. Likewise, if a bank account has a negative balance, it means we owe money rather than possess it. In business, a negative profit simply means a loss.
Many researchers have suggested that human beings are born with a natural number sense—a genetic ability to recognise changes in quantity. Many scientists also believe that numerous animals and birds can distinguish between one egg and several eggs, or between small and larger quantities.
One task for you
Take a sheet of paper and a pencil. Throughout one day, write down every occasion on which you use numbers—from the moment you wake up until you go to bed. I believe your list will be surprisingly long, because numbers are an inseparable part of the way we understand the world.
What we learned in this lesson
- When and where we use numbers.
- Counting and ordering numbers.
- Cardinal and ordinal numbers.
- The symbol for zero (0).
- Positive and negative numbers.
In our next lesson about numbers, we will learn how to compare them, explore their history, discover different numeral systems used throughout the world, examine the sacred and symbolic meanings of certain numbers, learn about the number π (pi), and see how numbers are named in different languages.
(Elena S. Lyubenova)
No comments:
Post a Comment