Compare and order the natural numbers

### 1.1. Knowledge to remember

#### 1.1.1. Compare natural numbers

**a) Of two natural numbers:**

– The number with more digits, the other number is larger. For example: 100 > 99.

The number with fewer digits is smaller. For example: 99 < 100

– If two numbers have the same digits, compare each pair of digits in the same row from left to right.

Such as :

- 29 869 and 30 005 both have five digits, in tens of thousands (tens of thousands) there are 2 < 3, so 29 869 < 30 005
- 25 136 and 23 894 both have five digits, the digits in the tens of thousands are 1, in the thousands there are 5 > 3, so: 25 136 > 23 894

– If two numbers have all pairs of digits in each row equal, then the two numbers are equal.

*It is always possible to compare two natural numbers, that is, to determine whether one is greater than, or less than, or equal to the other.*

**b) Comments:**

– In the natural number 0; first; 2; 3; 4; 5; 6; 7; 8; 9; … : The number before is smaller than the number after (eg: 8 < 9), the number after is larger than the number before (eg: 9 > 8).

– On the number ray: The number closer to 0 is the smaller number (eg: 1 < 5 ; 2 < 5; ...), clearly the number 0 is the smallest natural number : 0 < 1; 0 < 2; ... . Numbers farther from the origin 0 are larger numbers (eg: 12 > 11 ; 12 > 10 ; …).

#### 1.1.2. Order the natural numbers

Since natural numbers can be compared, it is possible to order the natural numbers from smallest to largest or vice versa.

**For example : **With the numbers 7698; 7968;7896;7869 can :

– Order from smallest to largest: 7698; 7869; 7896; 7968

– In order from largest to smallest: 7968; 7896; 7869; 7698

### 1.2. Textbook exercise solution page 22

**Lesson 1: **Put >, <, = in the dot

1234 … 999 35 784 … 35 790

8754 … 87 540 92 501 … 92 410

39 680 … 39000 + 680 17 600 … 17000 + 600

**Solution guide:**

Of two natural numbers:

- The number with more digits is larger. The number with fewer digits is smaller.
- If two numbers have equal digits, compare each pair of digits in the same row from left to right.
- If two numbers have all the same pairs of digits in each row, then the two numbers are equal.

1234 > 999 35 784 < 35 790

8754 < 87 540 92 501 > 92 410

39 680 = 39000 + 680 17 600 = 17000 + 600

**Lesson 2:** Write the following numbers in order from smallest to largest

a) 8316 ; 8136 ; 8361.

b) 5724 ; 5742 ; 5740.

c) 64 831 ; 64 813 ; 63 841.

__Solution guide:__

- Compare the given numbers, then write the numbers in order from smallest to largest.

a) We have: 8136 < 8316 < 8361.

So the numbers written in order from smallest to largest are: 8136 ; 8316 ; 8361.

b) We have: 5724 < 5740 < 5742.

So the numbers written in order from smallest to largest are: 5724 ; 5740 ; 5742.

c) We have: 1890 < 1945 < 1954 < 1969.

So the numbers written in order from smallest to largest are: 1890; 1945 ; 1954 ; 1969.

**Lesson 3:** Write the following numbers in order from largest to smallest

a) 1942 ; 1978 ; 1952 ; 1984.

b) 1890 ; 1945 ; 1969 ; 1954.

__Solution guide:__

- Compare the given numbers, then write the numbers in order from largest to smallest.

a) We have: 1984 > 1978 > 1952 > 1942.

So the numbers written in order from largest to smallest are: 1984 ; 1978 ; 1952 ; 1942.

b) We have: 1969 > 1954 > 1945 > 1890.

So the numbers written in order from largest to smallest are: 1969 ; 1954 ; 1945 ; 1890.

### 1.3. Exercise Solution Textbook Practice page 22

**Lesson 1:**

a) Write the smallest number: one digit, two digits; have three digits.

b) Write the largest number: one digit; have two digits; have three digits.

__Solution guide:__

- Based on the theory of the sequence of natural numbers to write the numbers according to the problem requirements.

a) The smallest one-digit, two-digit, three-digit number is 0 ; ten ; 100.

b) Write the largest single-digit, two-digit, and three-digit number as 9 ; 99 ; 999.

**Lesson 2:**

a) How many single-digit numbers are there?

b) How many two-digit numbers are there?

__Solution guide:__

- Method 1: Count the numbers that satisfy the requirements of the problem.
- Method 2: Apply the formula to find the number of terms of a series of equidistant numbers:

Number of terms = (last number – first number): the distance between two numbers + 1.

a) There are 10 digits where one digit is: 0; first; 2; 3; 4; 5; 6; 7; 8; 9.

b) The sequence of two-digit numbers is: 10; 11; twelfth; … 97; 98; 99.

The above sequence is a sequence of numbers that are equidistant, two consecutive numbers are more or less 1 unit apart.

The given sequence of numbers has terms of :

(99 – 10) : 1 + 1 = 90 (term)

So there are 90 two-digit numbers.

**Lesson 3:** Write the correct number in the blank

a) 85967 < 859 167 ; b) 42 037 > 482 037 ;

c) 609 608 < 609 60 ; d) 264 309 = 64 309.

__Solution guide:__

Of two natural numbers:

- The number with more digits is larger. The number with fewer digits is smaller.
- If two numbers have equal digits, compare each pair of digits in the same row from left to right.
- If two numbers have all the same pairs of digits in each row, then the two numbers are equal.

a) 859 067 < 859 167; b) 492 037 > 482 037;

c) 609 608 < 609 609; d) 264 309 = 264 309.

**Lesson 4: **Find the natural number x, know

a) x < 5 ; b) 2 < x < 5

*Attention:* It can be solved as follows, for example:

a) The natural numbers less than 5 are: 0; first; 2; 3; 4. So x is 0; first; 2; 3; 4.

__Solution guide:__

- Find the natural numbers less than 5, then find x.
- Find the naturals greater than 2 and less than 5, then find x.

a) The natural numbers less than 5 are 0; first; 2; 3; 4.

So x is: 0; first; 2; 3; 4.

b) The natural numbers greater than 2 and less than 5 are 3 and 4.

So x is: 3; 4.

**Lesson 5:** Find the round number of tens x, known: 68 < x < 92.

__Solution guide:__

- Find the round numbers greater than 68 and less than 92, from which we find x.
- The round numbers from smallest to largest are 10, respectively; 20 ; 30 ; 40 ; 50 ; … .

Round numbers greater than 68 and less than 92 are 70, 80, 90.

So x is: 70 ; 80 ; 90.

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