DECIMAL NUMBERS

Maybe you are the kind of person who thinks that you do not need decimals at all. You are wrong, because in our daily lives as non-mathematicians, we use decimals very actively. Let us remember where we use them.

In many languages, the concept of a fraction is expressed through words connected with breaking or dividing. In Latin, the word fractura is derived from frango, meaning “to break”.

Decimal fractions, in turn, became widely used in the late sixteenth century after the publication of the book De Thiende (The Tenth) by the Flemish engineer and mathematician Simon Stevin in 1585. Mathematicians also wrote about converting common fractions into decimals and vice versa during the seventeenth century.

The ways in which decimals have been written have also differed noticeably over the centuries. The fraction bar was probably borrowed from Arab mathematicians. However, even in the middle of the seventeenth century, some mathematicians still did not use it. The decimal point was introduced in the sixteenth century and was later popularised by John Napier.

There are two types of parts of whole numbers that we use actively: decimals and fractions. What are they? I suppose many people suspect that these are all those parts of numbers that lie between whole numbers. For example, between the whole numbers 1 and 2, some decimals are 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0. These examples represent a whole number and part of another whole. These parts are usually separated from the whole number by a decimal point or, in some countries, a decimal comma. Parts of whole numbers can be represented by common fractions or decimals.

How do we read decimals? For example:

0.5 is read as “zero point five” or “five tenths”.

0.37 is read as “zero point three seven” or “thirty-seven hundredths”.

0.000009 is read as “zero point zero zero zero zero zero nine” or “nine millionths”.

2.05 is read as “two point zero five” or “two and five hundredths”.

41.057 is read as “forty-one point zero five seven” or “forty-one and fifty-seven thousandths”.

I think there is no one who does not use decimal fractions on almost every day of their life. I believe everyone needs to use numbers that are not whole in certain situations. Almost everyone uses halves, quarters and thirds. Perhaps not everyone has to write them as numbers, and hardly anyone is so diligent and precise that they use all the digits after the decimal point with complete accuracy.

In which situations in everyday life can we use decimals? We use them whenever we talk about money, for example. If we have to mention fifty pence, we can write £0.50. When we measure height or distance, we also use decimals to show that a measurement is not expressed only in whole numbers. For example, a table may be 1.56 metres high, or a person may be 1.62 metres tall. When we follow cooking recipes, the weights of ingredients can also be written as decimals if they are not given in whole kilograms. We may also use decimals when recording temperature.

What mathematical operations can we perform with decimal fractions? We can add, subtract, multiply and divide decimals by one another and by whole numbers. I suppose this is where many of us may experience difficulties, but in our daily lives we can arrange decimals in columns to make the calculations easier, or we can use a calculator.

The rules for comparing decimals are as follows:

• Of two decimals, the greater one is the one with the greater whole-number part.

• If the whole-number parts are equal, we compare the tenths. The decimal with the greater number of tenths is larger.

• If the tenths are equal, we compare the hundredths, then the thousandths, and so on.

If the whole-number parts and all the decimal digits are equal, then the decimals are equal.

Example: Which number is greater: 3 or 1.24?

Because the whole-number parts satisfy 3 > 1, it follows that 3 > 1.24.

To add two decimals, we add units to units, tenths to tenths, hundredths to hundredths and thousandths to thousandths. If a decimal does not contain a digit in one of these places, we can write a zero.

The basic rules for adding decimals are:

• Equalise the number of decimal places by adding zeros where necessary.

• Write the numbers one below the other so that the decimal points are aligned.

• Add the numbers in the same way as whole numbers.

• Place the decimal point in the answer directly below the other decimal points.

For example:

0.356 + 43.7 = 44.056

We subtract decimals in a similar way.

To multiply two decimals, we follow these rules:

• Write the numbers one below the other.

• Ignore the decimal points and multiply the numbers as though they were whole numbers.

• Count the total number of decimal places in both factors.

• Place the decimal point in the answer so that it has the same total number of decimal places.

Examples:

12.25 × 10 = 122.5

2.327 × 100 = 232.7

1.3678 × 1,000 = 1,367.8

I suppose you agree that decimals are not always easy to work with because you have to remember many details if you want to be accurate. It is all about place value. In addition, if you have to multiply or divide them quickly, it can be difficult to do mentally. That is why we sometimes round decimal numbers to the nearest whole number, because whole numbers are easier to work with and remember.

For example, 0.6 is rounded to 1, while 1.2 is rounded to 1. We follow the familiar rule by looking at the digit immediately after the place to which we are rounding. If this digit is 5, 6, 7, 8 or 9, we round up. For example, 38 rounded to the nearest ten is 40. If the digit is 0, 1, 2, 3 or 4, we round down. For example, 33 rounded to the nearest ten is 30.

It is not an exaggeration to say that whenever we work with money and measurements, we often round decimals.

Are there decimal numbers in nature and in human life? Nature is all about wholeness. If the idea of a part of something appears in nature, it is mainly connected with the way human beings perceive the life around them. In our lives, we very often use the expression “part of me”, meaning that we have reached a point at which our wholeness cannot be achieved, at least in relation to certain things. It is usually a moment of imbalance and a lack of inner harmony.

Part of me understands this, but part of me does not. How can we measure this fraction of our brain and mind? What part of the whole is it? It is difficult to say. Or let us consider another expression: “I left part of my heart there.” Can we measure this part with a decimal number?

Mathematics is powerless in many situations, especially when human language reveals its creative and expressive power. From a spiritual point of view, we should aim to achieve and maintain wholeness in our mind, body and heart. All three dimensions need to be in the same place at the same time. Then there is peace, calmness and love for every single part of our whole being.

Here are some activities for you:

Activity 1: Can you make a list of all the situations in which you have had to use decimals today? Specify which mathematical operations you performed with them.

Activity 2: Can you write down some decimal numbers that lie between 0 and 10? Do not forget about place value.