Summary of Decimal System

Decimal System | Traditional Summary

Contextualization

The numbering system we use daily is called the 'Decimal System'. This system consists of ten different symbols, which are the digits from 0 to 9. Each position that a digit occupies in a number has a specific value, determined by the corresponding power of 10. For example, in the number 345, the digit 4 is in the tens position, meaning it represents 4 times 10, or 40. This system is widely used in various everyday situations, such as in financial transactions, measurements, and counting objects.

An interesting fact about the Decimal System is that it was developed by Indian and Arab mathematicians and spread around the world through trade and cultural exchanges. Today, it is the most widely used numbering system globally and is essential for science, technology, commerce, and the organization of modern society. Understanding the Decimal System and its practical applications is crucial for performing accurate calculations and making informed decisions in various areas of life.

Concept of the Decimal System

The Decimal System is a numbering system based on 10 different symbols, called digits. These digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These symbols can be combined in various ways to represent any number. The base 10 of the Decimal System means that each position of a digit in a number has a value that is a power of 10.

Each position in a decimal number represents a power of 10. For example, in the number 345, the digit 5 is in the units position (10^0), the 4 is in the tens position (10^1), and the 3 is in the hundreds position (10^2). Therefore, the number 345 is equal to 3100 + 410 + 5*1.

The decimal comma is used to separate the integer part from the decimal part of a number. For example, in the number 12.34, the integer part is 12 and the decimal part is 34 hundredths (0.34). The position to the right of the decimal comma represents fractions of 10, such as tenths, hundredths, thousandths, and so forth.

• The Decimal System uses 10 different symbols: 0 to 9.

• Each position of a digit in a number represents a power of 10.

• The decimal comma separates the integer part from the decimal part of a number.

Positions and Values

Each digit in a decimal number has a specific positional value. This value is determined by the position of the digit and the power of 10 corresponding to that position. For example, in the number 572.38, the digit 5 is in the hundreds position, 7 in the tens position, 2 in the units position, 3 in the tenths position, and 8 in the hundredths position.

The units position (10^0) is the first to the right of the decimal comma. As we move to the right, each position represents a fraction of 10: tenths (10^-1), hundredths (10^-2), thousandths (10^-3), etc. Similarly, as we move to the left, each position represents a larger power of 10: tens (10^1), hundreds (10^2), thousands (10^3), etc.

Understanding positional values is essential for performing arithmetic operations correctly, such as addition, subtraction, multiplication, and division, as each digit must be manipulated according to its positional value.

• The positional value of a digit is determined by its position and the corresponding power of 10.

• The units position is the first to the right of the decimal comma.

• Positional values are fundamental for correct arithmetic operations.

Reading and Writing Decimal Numbers

To read decimal numbers correctly, it is important to identify the integer part and the decimal part of the number. For example, the number 3.45 is read as 'three whole and forty-five hundredths'. The decimal part starts after the comma and is read as a fraction of 10.

When writing decimal numbers, attention must be paid to the decimal comma, which separates the integer part from the decimal part. The number 0.75, for example, represents 75 hundredths or 3/4. Precision in writing decimal numbers is crucial to avoid errors in calculations and measurements.

Converting between fractions and decimals is a useful skill. For example, the fraction 1/2 can be written as 0.5 in decimal format. Similarly, the fraction 3/4 can be represented as 0.75. This conversion facilitates comparison and the performance of arithmetic operations.

• Identifying the integer part and the decimal part is essential for the correct reading of decimal numbers.

• The decimal comma separates the integer part from the decimal part in the writing of decimal numbers.

• Converting between fractions and decimals facilitates comparison and arithmetic operations.

Comparison of Decimal Numbers

To compare decimal numbers, it is necessary to analyze each decimal place, starting from the left. If the numbers have the same integer part, the comparison must continue with the decimal part, digit by digit. For example, to compare 5.67 and 5.678, one starts with the tenths place, then the hundredths, and finally the thousandths.

Correct comparison of decimal numbers is important in various practical situations, such as in financial transactions, where precision is crucial. An error in comparison can lead to incorrect results and, consequently, wrong decisions.

Practical exercises in comparing decimal numbers help consolidate this knowledge. For example, comparing 3.45 and 3.450 may seem confusing, but both numbers are equivalent since zeros to the right in the decimal part do not change the value of the number.

• Comparing decimal numbers involves analyzing each decimal place from left to right.

• Precision in comparison is crucial in financial transactions and other practical situations.

• Practical exercises help consolidate knowledge about comparing decimal numbers.

To Remember

• Decimal System: A numbering system that uses 10 symbols, from 0 to 9, and base 10 to determine the positional value of the digits.

• Positional Value: The value of a digit in a number, determined by the position of the digit and the corresponding power of 10.

• Decimal Comma: A symbol that separates the integer part from the decimal part in a number.

• Integer Part: The part of the number that is to the left of the decimal comma.

• Decimal Part: The part of the number that is to the right of the decimal comma and represents fractions of 10.

• Comparison of Decimal Numbers: The process of analyzing each decimal place of two numbers to determine which is greater, lesser, or if they are equal.

Conclusion

The Decimal System is fundamental for understanding numbers and mathematical operations in everyday life. It uses ten symbols, from 0 to 9, and each position of a digit in a number has a specific value determined by the corresponding power of 10. Understanding positional values is essential for correctly performing arithmetic operations and for accuracy in calculations and measurements.

Reading and writing decimal numbers correctly is a crucial skill that facilitates accurate calculations and comprehension of practical situations, such as in financial transactions and measurements. The comparison of decimal numbers, analyzing each decimal place from left to right, is vital to ensure precision and avoid errors in various contexts.

Knowledge of the Decimal System is widely applied in various fields, such as science, technology, and commerce. Precision in reading, writing, and comparing decimal numbers is essential for making informed decisions and performing correct calculations in everyday situations. Exploring more about the Decimal System will help students develop solid and applicable mathematical skills.

Study Tips

• Practice reading and writing different decimal numbers, identifying the integer part and the decimal part to ensure precision.

• Carry out exercises comparing decimal numbers, analyzing each decimal place to develop the skill to determine which are larger or smaller.

• Explore practical applications of the Decimal System in daily life, such as in financial transactions and measurements, to understand the relevance and importance of this knowledge.