What is a decimal? Calculations with decimals

Along with rational numbers , real numbers , prime numbers , etc., decimals are a familiar type of number and are commonly used in mathematics and life. Quantrimang.com would like to introduce to you an overview of decimals including concepts, structures, how to write decimals, how to read, how to add, subtract, multiply, and divide decimals, etc. We invite you to learn so that you can apply them to your studies as well as in life.

Decimal

• What is a decimal?
  • Decimal fraction
  • Decimal
• How to write decimal fractions as decimal numbers and vice versa
• Compare two decimals
  • How to compare two positive decimal numbers
• Addition and subtraction of decimals
  • Rules for adding and subtracting two positive decimal numbers
• Multiplication and division of decimal numbers
  • Rules for multiplying two positive decimal numbers
  • Rules for dividing two positive decimals
• Decimal classification
  • 1. Finite decimal numbers
  • 2. Infinite recurring decimals
• How to round decimals

What is a decimal?

Decimal fraction

Decimal fractions are fractions whose denominators are powers of 10 and whose numerators are whole numbers.

For example:

Decimal

- A decimal number consists of two parts: the integer part is written to the left of the "," sign; the decimal part is written to the right of the "," sign.

- After the ",":

• First digit: tenths place
• Second digit: hundredths place
• Third digit: thousandths place; ...

- Every decimal fraction is written as a decimal number and vice versa.

For example: Identify the integer part, the decimal part and state how to read the following decimal numbers.

Instruct

a) Integer part: −812; decimal part: 603 Read as: Negative eight hundred twelve point six hundred and three.

b) Integer part: 3474; decimal part: 1

Read as: Three thousand four hundred seventy four point one.

c) Integer part: −99; decimal part: 15 Read as: Negative ninety-nine point fifteen.

d) Integer part: −35; decimal part: 703 Read as: Negative thirty-five point seven hundred and three.

For example: Write the following decimal numbers knowing:

a) The negative decimal number has the integer part being the largest 2-digit number divisible by 5, and the decimal part being the smallest 3-digit number divisible by 3.

b) The largest positive decimal number with an integer part of 3 digits, the decimal part including the tenths place is 8.

c) A negative decimal number has the integer part being the smallest 3-digit number divisible by 9, the decimal part including the tenths place being 1 and the hundredths place being the smallest number divisible by 5 and not divisible by 2.

Instruct

How to write decimal fractions as decimal numbers and vice versa

Method

• To write a decimal fraction as a decimal number, we count how many zeros there are in the denominator, then use the “,” sign in the numerator so that the number of digits after the decimal point is equal to the number of zeros counted. If the numerator does not have enough digits, we add zeros to the left of it.
• To write a decimal number as a decimal fraction, we count the number of digits after the decimal point, set the fraction so that the numerator is the decimal part (no decimal point), the denominator is a power of 10 with the exponent equal to the number of digits after the decimal point.

Note: Fractions whose denominators have no prime factors other than 2 and 5 can be written as both decimal fractions and decimal numbers.

Example : Convert the following decimal fractions (mixed numbers) into decimals and then find their opposites:

Instruct

a) ; The opposite number is 0.01

b) ; The opposite number is 5.67

c) ; The opposite number is −9.5

d) ; The opposite number is 2.02.

For example: Write the following decimal numbers as decimal fractions and then find their opposites:

Instruct

a) ; the opposite number is:

b) ; the opposite number is:

c) ; the opposite number is:

d) ; the opposite number is:

Compare two decimals

Principle

Negative decimals are less than 0 and positive decimals are greater than 0.

If are two positive decimal numbers and then

How to compare two positive decimal numbers

+ Compare the integer part of the two positive decimal numbers. The decimal number with the larger integer part is larger.

+ If the two positive decimal numbers have the same integer part, we continue to compare each pair of digits in the same row (after the “,” sign) from left to right until the first pair of different digits appears. In that pair of different digits, the larger digit is the larger decimal number containing that digit.

For example: Compare:

Instruct

a) 74.25 > 74.201

b) 940.13 940.15

Example: Arrange the following decimal numbers in ascending order:

Instruct

The decimal numbers in ascending order are:

Addition and subtraction of decimals

Rules for adding and subtracting two positive decimal numbers

Step 1: Write this number below the other number so that the digits in the same row are aligned with each other, and the "," are aligned with each other.

Step 2: Perform addition and subtraction like adding and subtracting natural numbers.

Step 3: Write the “,” in the result in the same column as the “,” written above.

• Add two negative decimals: with
• Add two decimal numbers with different signs:
  • If
  • If

Subtraction of two decimal numbers is reduced to addition with the opposite number.

Multiplication and division of decimal numbers

Rules for multiplying two positive decimal numbers

Step 1: Remove the “,” and multiply like multiplying two natural numbers.

Step 2: Count how many digits the decimal part of both factors has, then use the "," symbol to separate the product into that many digits from left to right.

Rules for dividing two positive decimals

Step 1: Count how many digits are in the decimal part of the divisor, then move the “,” sign in the dividend that many digits to the right.

Note: When moving the "," sign in the dividend to the right but there are not enough digits, we see that there are enough missing digits, then add that many zeros.

Step 2: Remove the “,” in the divisor and then perform the division as dividing a decimal by a natural number.

• The product and quotient of two decimal numbers with the same sign is always a positive number.
• The product and quotient of two decimal numbers with different signs is always a negative number.

- Decimal numbers have all the properties: commutative, associative, distributive properties of multiplication over addition, addition by 0 and multiplication by 1.

For example: Calculate:

Instruct

Decimal classification

1. Finite decimal numbers

If a fraction is reduced to its positive denominator and has no prime factors other than 2 and 5, then the fraction is written as a terminating decimal.

For example :

2. Infinite recurring decimals

If a decimal is reduced to a positive denominator and the denominator has prime factors other than 2 and 5, then the fraction can be written as an infinite recurring decimal.

For example :

- Every rational number is represented by a finite recurring decimal or an infinite recurring decimal.

- Each finite recurring decimal represents a rational number.

For example: Explain why fractions can be written as terminating decimals? Write them in that form.

Instruct

Fractions can be written as terminating decimals because their denominators have prime factors 2 and 5.

For example: Explain why fractions can be written as non-terminating recurring decimals? Rewrite them as non-terminating recurring decimals.

Instruct

The fractions, whose denominators have prime factors other than 2 and 5, are written as infinite recurring decimals.

How to round decimals

To round a positive decimal number to a certain place, do the following:

- For rounded digits:

• Keep if the digit to the right is less than 5.
• Increase by 1 if the digit to the right is greater than or equal to 5.

- For the digits after the rounding place:

• Omit if in decimal part
• Replace with zeros if in the integer part.

- When rounding a number to a certain place, the rounded result has an accuracy of half the rounded place unit.

- To round a decimal number with a given precision, we can determine the rounding row through the following data table:

Example: Find the fifth decimal place of the number and round the number to the fifth decimal place.

Instruct

We have:

=> The fifth decimal digit of that number is 1.

Rounding the number to the 5th decimal place we get:

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Hopefully the above article has helped you understand decimals as well as decimal operations.

In addition to decimals, you can learn more about other common types of numbers such as fractions , integers ...