Recording natural numbers is a common form of math in the 6th grade math program in middle school. So what is a natural number? What types of math are there to write natural numbers? How to solve grade 6 math with natural numbers?… In the content of the article below, Tip.edu.vn will help you synthesize knowledge on the above topic, let’s find out!.
What is a natural number?
There are many different definitions of natural numbers, such as
- The known natural numbers are all the numbers ( 0,1,2,3,4,5,…)
- The natural number in its main concept is the set of all non-negative integers.
Natural numbers are used for two main purposes:
- Used to count. Example: There are ( 5 ) people in a room
- Used to sort the hierarchy. For example: Vietnam is the country with the population ranked ( 15 ) in the world.
Natural number representation
To write natural numbers we use ten digits: ( 0.1,2,3,4,5,6,7,8,9 )
Natural numbers can include one digit, two digits, three digits, ….
Attention: The first digit of a natural number is never a number ( 0 )
Eg:
- ( 1 ) is a single-digit natural number.
- ( 25 ) is a two-digit natural number.
- ( 141 ) is a three-digit natural number.
When writing natural numbers with five or more digits, it is common to write each group of three digits separately from right to left for readability. For example :(23;213)
In the evolution of natural numbers:
- The last digit that appears is called the units digit.
- The digit immediately to the left of the units digit is called the tens digit.
- The digit to the right of the tens digit is called the hundreds digit.
Just like that, we have the digits of thousands, digits of tens of thousands, digits of hundreds of thousands, etc.
For every ( 10 ) units in a row, it forms ( 1 ) units in the adjacent row to the left of that row.
For example, the number ( 2019 ) has :
- The unit digit is ( 9 )
- The tens digit is ( 1 )
- The hundreds digit is ( 0 )
- Thousands digit ( 2 )
The way to write numbers as above is how to write numbers in the decimal system. There are also other ways of writing numbers, such as Roman numerals:
Mathematical forms for recording natural numbers
Form 1: Write natural numbers according to digits, rows
To solve this problem, we need to pay attention to distinguish tens from tens digit, hundreds with hundreds digit, etc. At the same time, we must understand the transformation rule: ( 10 ) units in a row, create to ( 1 ) units in the adjacent row to the left of that row
Example 1: (Exercise 11 math grade 6 write natural numbers page 10)
a) Write a natural number whose tens is ( 135 ) , the units digit is ( 7 )
b) Fill in the table below with the missing cells:
Solution:
a, A natural number consisting of ( 135 ) tens and ( 7 ) whose units are ( 1357 )
b, To find the number of tens we remove the last digit of the given number
Similarly, to find the number of hundred, we remove the last two digits of the given number
So we have the following result:
Example 2: (Exercise 13 math grade 6 natural number writing lesson page 10)
a) Write the smallest four-digit natural number.
b) Write the smallest natural number with four distinct digits.
Solution:
a) The smallest natural number has four digits, the digits of that number must have the smallest possible value
Since the first digit must be different from ( 0 ), so for the smallest number, the first digit must be ( 1 )
The next three smallest digits are the number ( 0 )
So the smallest natural number with four digits is ( 1000 )
b) A number with four distinct digits is the smallest then:
Its first digit must be the smallest number other than ( 0 ) i.e. the number ( 1 )
The next digit must be the smallest number other than ( 1 ) i.e. ( 0 )
The next digit must be the smallest number other than ( 0 ) and ( 1 ) i.e. the number ( 2 )
The unit digit must be the smallest other than ( 0,1,2 ) ie the number ( 3 )
So the smallest natural number with four distinct digits is ( 1023 )
Form 2: Write natural numbers from given digits
Problem: From the digits ( a,b,c neq 0 ) write all the different three-digit numbers that are made up of those three digits
Solution:
Choose ( a ) as the hundreds digit, then we can write two numbers: (overline{abc},overline{acb})
Choose ( b ) as the hundreds digit, then we can write two numbers: (overline{bac},overline{bca})
Choose ( c ) as the hundreds digit, then we can write two numbers: (overline{cab},overline{cba})
Thus, in total, we can write six numbers that satisfy the requirements of the problem
Do the same for four-digit, five-digit numbers
Attention: If the given digits contain the number ( 0 ), it should be noted that the number ( 0 ) never comes first.
Eg: (Exercise 14, math 6, write natural numbers on page 10).
Using three digits ( 0, 1, 2 ) write all three-digit natural numbers whose digits are different.
Solution:
The hundreds digit must be different from ( 0 ) so the hundreds digit can be ( 1 ) or ( 1 ) .
If the hundreds digit is ( 1 ) we can write two numbers as ( 102,120 )
If the hundreds digit is ( 2 ) we can write two numbers as ( 201,210 )
So with three digits ( 0,1,2 ) we can write all four three-digit natural numbers whose different digits are ( 102 ; 120 ; 201; 210 )
Form 3: Find the number of natural numbers that satisfy the requirements of the problem
To solve this problem, we use the following formula for counting equidistant numbers:
The number of terms from ( a ) to ( b ) where two adjacent numbers are separated by ( c ) units is : (frac{ba}{c}+1)
In other words, to count the number of terms in a sequence of equidistant numbers, we subtract the first number from the last number, then divide the difference by the distance between two consecutive numbers, then add ( 1 ) units.
In addition, to calculate the sum of all terms in a series of equally spaced numbers, we have the formula: (frac{(a+b)times n}{2})
Where: ( a,b ) are the first and last two terms
( n ) is the number of terms in the sequence calculated by the formula (n= frac{ba}{c}+1)
Eg:
a, How many even four-digit numbers are there in all?
b, Sum all two-digit odd numbers
Solution:
a, The smallest even four-digit number is ( 1000 )
The largest four-digit even number is ( 9998 )
The distance between two consecutive even numbers is ( 2 ) units
So, applying the formula, we get the number of even four-digit numbers: (frac{9998-1000}{2}+1= 4500)
b, The smallest two-digit odd number is ( 11 )
The largest two-digit odd number is ( 99 )
The distance between two consecutive odd numbers is ( 2 ) units
So the number of odd two-digit numbers is: (frac{99-11}{2}+1=45)
So the sum of all two-digit odd numbers is: (frac{(99+11) times 45}{2}= 2475)
Form 4: Count the number of digits used in the number sequence
To solve problems of this type, we first need to master the formula for calculating the number of terms in the sequence of numbers as above.
We consider the digits in each row of units, tens, … see how many times the digit to find is used. Then add them all together to get the result
Eg:
How many digits are needed to write all three-digit numbers ( 1 )
Solution:
- Three-digit numbers containing ( 1 ) in the units row are : ( 101,111,121,…991 )
Each number in the above sequence is separated by ( 10 ) units, so the number of digits ( 1 ) used in the main units row is equal to the number of three-digit numbers with the units row of ( 1 ) and equals : (frac{991-101}{10}+1=90) number
- Three-digit numbers that contain ( 1 ) in the tens place are:
( 110,111,…119 ) has ( 10 ) numbers
( 210,211,…219 ) has ( 10 numbers
….
[latex] 910 , 911, …919 ) have ( 10 ) numbers
Thus there are all ( 90 ) three digit numbers containing the digit ( 1 ) in the tens place. In other words, the digit ( 1 ) is used ( 90 ) to write tens digits
- Three-digit numbers containing ( 1 ) in the hundreds are : ( 100,101,…199 )
Each number in the sequence above is separated by ( 1 ) units, so the number of digits ( 1 ) used in the hundreds is exactly the same as the number of three-digit numbers in the hundreds is ( 1 ) and equals : (frac{199-100}{1}+1=100) number
So to write all three digit numbers we need to use ( 90+90+100 =280 ) digits ( 1 )
Attention: We can count in another way as follows:
Three-digit numbers of the form (overline{abc})
If ( c=1 ) then ( a ) has ( 9 ) the choice is ( 1,2,…,9 ) and ( b ) has ( 10 ) the choice is ( 0.1,…,9 )
Thus there are ( 9 times 10 =90 ) three digit numbers whose units are ( 1 )
Similarly, there are ( 9 times 10 = 90 ) three digit numbers where tens equals ( 1 )
There are ( 10 times 10 =100 ) three digit numbers where hundreds equals ( 1 )
Form 5: Math problems on reading and writing Roman numerals
To do math problems with Roman numerals, first we need to know the basic Roman numeral symbols:
( I = 1 )
( V = 5 )
( X = 10 )
( L = 50 )
( C = 100 )
( D=500 )
( M = 1000 )
Some rules for writing Roman numerals:
- The added digit on the right is an addition (smaller than the original digit on the left) and must never be more than three times the number. Eg :
The number ( 8 ) is ( VIII ) but the number ( 9 ) is ( IX ) not ( VIIII )
( 2238 = 2000 + 200 + 30 + 8 = MMCCXXXVIII )
- The numbers written on the left are usually subtracted, that is, subtracting the original number from the left will give the value of the calculation (provided that the left digit is smaller than the right digit). Eg :
The number ( 4 ) is ( IV )
The number ( 90 ) is ( XC )
( MCMXCIX = M + CM + XC + IX = 1000+900+90+9=1999 )
Eg: (Exercise 15 math grade 6, volume 1, write natural numbers)
a) Read the following Roman numerals: ( XIV ; XXVI )
b) Write the following numbers in Roman numerals: ( 17 ; 25 )
c) Have nine matchsticks arranged as shown in figure 8. Move one match to get the correct result.
Solution:
a, We have:
( XIV = X + IV =10 + 4 =14 )
So ( XIV ) reads as ” fourteen “
( XXVI = XX + VI = 20+6 =26 )
So ( XXVI ) reads as “twenty-six”.
b, We have:
( 17= 10+ 7 = X+VII= XVII )
( 25 = 20 +5 = XX+ V= XXV )
c, We can move in different ways:
- Method 1: ( VI = V -I ) changes to ( V = VI -I ) or ( 5=6-1 )
- Method 2: ( VI = V -I ) changes to ( IV = V -I ) or ( 4=5-1 )
- Method 3: ( VI = V -I ) changes to ( VI – V = I ) or ( 6=5-1 )
Exercises on writing natural numbers grade 6
Here are some exercises with answers for you to practice on your own:
Lesson 1: (grade 6 math, write natural numbers, exercise 12)
Write the set of digits of the number ( 2000 )
Answer: ({2;0})
Lesson 2:
Find four consecutive natural numbers whose sum is the greatest three-digit even number
Answer: ( 248,249,250,251 )
Lesson 3:
How many three-digit natural numbers are there in all that contain the digit ( 3 )
Answer 🙁 352 ) number
Lesson 4:
Given five digits ( 1,3,5,0.4 )
a) Write the smallest five-digit natural number that can be formed from the above five digits
b, Write the largest five-digit natural number formed from the above five digits
Answer: a, ( 10345 )
b, ( 54310 )
Lesson 5:
Write the following numbers in Roman numeral form:
a, ( 145 )
b, ( 2019 )
Answer: a, ( CXLV )
b, ( MMXIX )
The above article of Tip.edu.vn has helped you to synthesize theory as well as exercises on mathematical forms of recording natural numbers in the 6th grade program. Hopefully the knowledge in the article will help you in the learning process. and research on the topic of recording natural numbers. Good luck with your studies!.
Please refer to the lecture on Recording Natural Numbers below:
(Source: www.youtube.com)
ly-thuyet-ghi-so-tu-nhien-toan-lop-6
See more:
- Transformation rule: Synthesize Theory and Basic Maths
- Topics on subtraction and division: Basic math forms and Exercises
- The topic of analyzing a number to make prime factors: Theory and Exercises
- Divisibility of a Sum: Basic Math Types and Advanced Exercises
Xem thêm nhiều bài viết hay về Hỏi Đáp Toán Học