Numbers beyond 9999
Understanding 5-Digit and 6-Digit Numbers
1. What is a 5-Digit Number?
A 5-digit number is a number that has 5 digits in it, and its value lies between 10,000 and 99,999. Here’s how it works:
– Smallest 5-digit number: 10,000
– Largest 5-digit number: 99,999
Place Value of 5-Digit Number
Each digit has a place value. The place values from right to left in a 5-digit number are:
For example, in the number 43,721:
– 4 is in the Ten Thousands place (40,000),
– 3 is in the Thousands place (3,000),
– 7 is in the Hundreds place (700),
– 2 is in the Tens place (20),
– 1 is in the Ones place (1).
So, the number can be written as:
40,000 + 3,000 + 700 + 20 + 1 = 43,721
2. What is a 6-Digit Number?
A 6-digit number is a number that has 6 digits, and its value lies between 1,00,000 and 9,99,999. Here’s how it works:
– Smallest 6-digit number: 1,00,000
– Largest 6-digit number: 9,99,999
Place Value of 6-Digit Number
The place values from right to left in a 6-digit number are:
For example, in the number 5,34,217:
– 5 is in the Lakhs place (5,00,000),
– 3 is in the Ten Thousands place (30,000),
– 4 is in the Thousands place (4,000),
– 2 is in the Hundreds place (200),
– 1 is in the Tens place (10),
– 7 is in the Ones place (7).
So, the number can be written as:
5,00,000 + 30,000 + 4,000 + 200 + 10 + 7 = 5,34,217
Comparing 5-Digit and 6-Digit Numbers
– 5-Digit numbers range from 10,000 to 99,999.
– 6-Digit numbers range from 1,00,000 to 9,99,999.
– As the number of digits increases, the value of the number increases.
Number system
The number system is a way to understand and work with numbers, focusing on place value, types of numbers, and how to use them in basic operations.
1. Digits and Numbers
– Digits are the basic symbols used to represent numbers. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
– Numbers are made by combining digits. For example, 23 is a number made from the digits 2 and 3.
2. Place Value
– The place value of a digit depends on its position in a number. From right to left, the places are ones, tens, hundreds, thousands, and so on.
Example:
In the number 3,642:
– The digit 2 is in the ones place (value = 2).
– The digit 4 is in the tens place (value = 40).
– The digit 6 is in the hundreds place (value = 600).
– The digit 3 is in the thousands place (value = 3,000).
3. Types of Numbers
– Natural Numbers: These are counting numbers starting from 1 (1, 2, 3, 4, 5…).
– Whole Numbers: These include all natural numbers and 0 (0, 1, 2, 3…).
– Even Numbers: Numbers that can be divided by 2 (e.g., 2, 4, 6, 8…).
– Odd Numbers: Numbers that cannot be divided by 2 (e.g., 1, 3, 5, 7…).
4. Expanded Form
This is a way of breaking down a number to show the value of each digit.
Example:
– 5,381 in expanded form is:
– 5,000 + 300 + 80 + 1.
5. Comparing Numbers
When comparing two numbers, look at the digits from left to right:
– Start with the highest place value (thousands, hundreds, etc.).
– A bigger digit in the higher place value means the number is bigger.
Example:
– 4,521 is greater than 3,521 because 4 in the thousands place is greater than 3.
6. Rounding Numbers
Rounding means making a number simpler but keeping it close to its original value.
– If the digit you’re rounding is followed by 5 or higher, round up.
– If it’s less than 5, round down.
Example:
– Rounding 67 to the nearest ten: Look at the 7. Since it’s more than 5, round 67 to **70**.
7. Roman Numerals
– Roman numerals use letters to represent numbers. Some basic Roman numerals are:
– I = 1
– V = 5
– X = 10
– L = 50
– C = 100
Example: XIII = 10 + 1 + 1 + 1 = 13.
Face value and place value
1. Face Value
The face value of a digit is the value of the digit itself, no matter where it is in the number. It doesn’t change with its position.
For example:
– In the number 4,683, the face value of:
– 4 is 4.
– 6 is 6.
– 8 is 8.
– 3 is 3.
So, the face value of a digit is simply the number you see.
2. Place Value
The place value of a digit depends on its position in the number. The place value increases as you move from right to left, starting from the ones place, tens place, hundreds place, and so on. Each position represents a power of 10.
For example, in the number 4,683:
– The place value of the digit 3 is 3 ones or 3.
– The place value of the digit 8 is 8 tens or 80.
– The place value of the digit 6 is 6 hundreds or 600.
– The place value of the digit 4 is 4 thousands or 4,000.
Difference Between Face Value and Place Value
| Digit | Face Value | Place Value |
|---|---|---|
|
3 (in 4,683) |
3 |
3 ones = 3
|
|
8 (in 4,683) |
8 |
8 tens = 80
|
|
6 (in 4,683) |
6 |
6 hundreds = 600
|
|
4 (in 4,683) |
4 |
4 thousands = 4,000
|
So:
– The face value is the actual digit.
– The place value is the digit’s value based on its position in the number.
Example:
Let’s take another number 7,352:
– Face Value of 7 = 7, but the Place Value is 7,000 (because it is in the thousands place).
– Face Value of 3 = 3, but the Place Value is 300 (because it is in the hundreds place).
Expanded form and Standard form
1. Expanded Form for 5-Digit Numbers:
A 5-digit number has digits in the Ten Thousands, Thousands, Hundreds, Tens, and Ones places.
Example 1:
Take the number 45,627.
In expanded form, this number is written as:
45,627 = 40,000 + 5,000 + 600 + 20 + 7
Here’s how:
– 4 is in the ten thousands place, so it’s 40,000.
– 5 is in the thousands place, so it’s 5,000.
– 6 is in the hundreds place, so it’s 600.
– 2 is in the tens place, so it’s 20.
– 7 is in the ones place, so it’s 7.
In standard form, this is written as: 45,627.
2. Expanded Form for 6-Digit Numbers:
A 6-digit number has digits in the Hundred Thousands, Ten Thousands, Thousands, Hundreds, Tens, and Ones places.
Example 2:
Take the number 532,184.
In expanded form, this number is written as:
532,184 = 500,000 + 30,000 + 2,000 + 100 + 80 + 4
Here’s how:
– 5 is in the hundred thousands place, so it’s 500,000.
– 3 is in the ten thousands place, so it’s 30,000.
– 2 is in the thousands place, so it’s 2,000.
– 1 is in the hundreds place, so it’s 100.
– 8 is in the tens place, so it’s 80.
– 4 is in the ones place, so it’s 4.
In standard form, this is written as: 532,184.
Summary:
– Expanded form breaks down the number based on the place value of each digit.
– Standard form is the usual way we write the number.
Comparing numbers
Comparing 5-Digit Numbers:
A 5-digit number has five places: Ten-thousands, Thousands, Hundreds, Tens, and Ones.
For example: 34,521 has the digits 3, 4, 5, 2, and 1.
Comparing 6-Digit Numbers:
A 6-digit number has six places: Hundred-thousands, Ten-thousands, Thousands, Hundreds, Tens, and Ones.
For example: 245,123 has the digits 2, 4, 5, 1, 2, and 3.
Steps to Compare 5-Digit and 6-Digit Numbers
1. Count the number of digits:
– Always start by counting how many digits the numbers have.
– A 6-digit number is always greater than a 5-digit number.
– Example: Compare 45,612 and 245,612.
– 245,612 has 6 digits.
– 45,612 has 5 digits.
– Since 245,612 has more digits, it’s automatically the larger number.
2. If both numbers have the same number of digits:
If both numbers have 5 or both have 6 digits, compare them digit by digit starting from the left.
Example 1 (5-digit numbers):
Compare 52,738 and 52,653:
– First digit (Ten-thousands): Both have 5.
– Second digit (Thousands): Both have 2.
– Third digit (Hundreds): Both have 7.
– Fourth digit (Tens): Compare 3 and 6. Since 3 < 6, 52,738 < 52,653.
Example 2 (6-digit numbers):
Compare 721,945 and 729,432:
– First digit (Hundred-thousands): Both have 7.
– Second digit (Ten-thousands): Compare 2 and 2 (same).
– Third digit (Thousands): Compare 1 and 9. Since 1 < 9, 721,945 < 729,432.
3. Compare Large Numbers Quickly:
To compare larger numbers like 5-digit or 6-digit ones quickly, always focus on the **largest place value** first. If the digits in the largest place value are the same, then compare the next place value.
Practice Examples:
1. Compare 45,672 and 98,123.
– Both are 5-digit numbers.
– Compare the Ten-thousands place: 4 < 9 (so, 45,672 < 98,123).
2. Compare 734,289 and 623,456.
– Both are 6-digit numbers.
– Compare the Hundred-thousands place: 7 > 6 (so, 734,289 > 623,456).
3. Compare 789,345 and 89,123.
– 789,345 is a 6-digit number.
– 89,123 is a 5-digit number.
– Since 789,345 has more digits, it is automatically greater.
Odering numbers
Ordering numbers means arranging them in ascending or descending order. For 5-digit and 6-digit numbers, the process is simple if you follow these steps:
1. What is a 5-Digit Number?
A 5-digit number has 5 digits, like:
– 12,345
– 67,890
– 45,678
The smallest 5-digit number is 10,000 and the largest 5-digit number is 99,999.
2. What is a 6-Digit Number?
A 6-digit number has 6 digits, like:
– 123,456
– 678,901
– 456,789
The smallest 6-digit number is 100,000 and the largest 6-digit number is 999,999.
How to Order Numbers:
Ascending Order (Smallest to Largest)
1. Compare the number of digits: A number with fewer digits is always smaller. For example, all 5-digit numbers are smaller than 6-digit numbers.
2. Compare the first digits (place values):
– Start with the left-most (largest place) digit. Compare the digits one by one.
– If the first digit is the same, move to the next digit.
Example:
– Numbers: 45,678, 12,345, 678,901, 123,456
– Step 1: The 5-digit numbers are 45,678 and 12,345; the 6-digit numbers are 678,901 and 123,456.
– Step 2: Order them:
– First, compare the number of digits: 5-digit numbers come before 6-digit numbers.
– For the 5-digit numbers: 12,345 < 45,678
– For the 6-digit numbers: 123,456 < 678,901
Final Ascending Order: 12,345 < 45,678 < 123,456 < 678,901.
Descending Order (Largest to Smallest)
You do the opposite: start from the largest number and go down.
Example:
– Step 1: Compare the 6-digit numbers first, then the 5-digit numbers.
– Step 2: 678,901 > 123,456 > 45,678 > 12,345
Final Descending Order: 678,901 > 123,456 > 45,678 > 12,345.
Key Points:
– More digits = Bigger number.
– If the numbers have the same number of digits, compare from the left-most digit.
This process helps you easily arrange numbers in ascending or descending order!
Successor and Predecessor
1. What is a Successor?
– The successor of a number is the number that comes just after the given number.
– To find the successor of a number, add 1 to it.
Example:
– Successor of 45,678 (a 5-digit number) is 45,679.
– Successor of 123,456 (a 6-digit number) is 123,457.
2. What is a Predecessor?
– The predecessor of a number is the number that comes just before the given number.
– To find the predecessor of a number, subtract 1 from it.
Example:
– Predecessor of 45,678 (a 5-digit number) is 45,677.
– Predecessor of 123,456 (a 6-digit number) is 123,455.
Examples :
1. 54,321
Successor: 54,322 (add 1)
Predecessor: 54,320 (subtract 1)
2. 10,000
– Successor: 10,001
– Predecessor: 9,999 (This becomes a 4-digit number)
3. 99,999
– Successor: 100,000 (This becomes a 6-digit number)
– Predecessor: 99,998
4. 654,321
– Successor**: 654,322 (add 1)
– Predecessor: 654,320 (subtract 1)
Key Points:
– Successor: Add 1.
– Predecessor: Subtract 1.
– A number’s successor is always greater, and its predecessor is always smaller.
Forming numbers
Forming numbers means arranging digits to create different numbers. Let’s explore how you can form 5-digit and 6-digit numbers using different digits.
1. What is a 5-Digit Number?
– A 5-digit number has exactly 5 digits.
– The first digit cannot be zero because the number would then become a 4-digit number.
– Example of a 5-digit number: 34,567.
2. What is a 6-Digit Number?
– A 6-digit number has exactly 6 digits.
– Similar to 5-digit numbers, the first digit cannot be zero.
– Example of a 6-digit number: 234,567.
Forming a 5-Digit Number
You need five digits to form a 5-digit number. Let’s look at how this works:
Example: Form a 5-digit number using the digits 2, 4, 5, 7, 9.
– Step 1: Arrange these digits in any order to form a number.
– Example arrangement: 54,792.
– Step 2: Ensure there are exactly 5 digits in your number.
– 54,792 is a valid 5-digit number.
– Step 3: Make sure the first digit is not 0, as this would turn it into a 4-digit number.
Multiple 5-Digit Numbers:
By rearranging the digits, you can form different 5-digit numbers:
– 52,479
– 74,259
– 94,572
Using Repeated Digits:
You can use digits more than once if allowed. For example, using the digits 3, 4, 5, you can form:
– 44,533
– 55,345
– 33,554
Forming a 6-Digit Number
Now let’s move to 6-digit numbers.
Example: Form a 6-digit number using the digits 3, 5, 6, 7, 8, 9.
– Step 1: Arrange these digits in any order to form a number.
– Example arrangement: 96,5873.
– Step 2: Make sure there are exactly 6 digits.
– 96,5873 is a valid 6-digit number.
Multiple 6-Digit Numbers:
By rearranging the digits, you can form different 6-digit numbers:
– 97,8635
– 35,9867
– 68,7395
Using Repeated Digits:
You can also form 6-digit numbers with repeated digits. Using 2, 5, you could form:
– 555,522
– 225,555
Arranging Numbers in Ascending or Descending Order
When forming numbers, you can also arrange the digits in different ways:
– Ascending Order: Arrange digits from smallest to largest.
– Example for 5, 7, 9, 3, 2: 23,579.
– Descending Order: Arrange digits from largest to smallest.
– Example for 5, 7, 9, 3, 2: 97,532.
Key Points for Class 4:
– A 5-digit number always has 5 digits, and the first digit can’t be zero.
– A 6-digit number always has 6 digits, and the first digit can’t be zero.
– You can rearrange given digits in different ways to form many different numbers.
– You can form numbers in ascending order (smallest to largest) or descending order (largest to smallest).
Rounding of numbers
Rounding means changing a number to a nearby value that is easier to work with. We usually round numbers to the nearest 10, 100, 1,000, or even higher values to make calculations simpler.
Why Do We Round Numbers?
– To make large numbers easier to handle.
– To estimate results when exact values are not needed.
Basic Rounding Rules:
When rounding numbers, the key rule is to look at the digit to the right of the place value you are rounding to:
1. If the digit is 5 or more, round up.
2. If the digit is less than 5, round down.
1. Rounding to the Nearest 10:
When rounding to the nearest 10, look at the ones place.
– If the ones place is 5 or more, round up.
– If the ones place is less than 5, round down.
Examples:
– 43 rounds to 40 (since 3 is less than 5).
– 78 rounds to 80 (since 8 is 5 or more).
2. Rounding to the Nearest 100:
When rounding to the nearest 100, look at the tens place.
– If the tens place is 5 or more, round up.
– If the tens place is less than 5, round down.
Examples:
– 423 rounds to 400*(since 2 is less than 5).
– 678 rounds to 700 (since 7 is 5 or more).
– 350 rounds to 400 (since 5 means round up).
3. Rounding to the Nearest 1,000:
When rounding to the nearest 1,000, look at the hundreds place.
– If the hundreds place is 5 or more, round up.
– If the hundreds place is less than 5, round down.
Examples:
– 2,340 rounds to 2,000 (since 3 is less than 5).
– 5,678 rounds to 6,000 (since 6 is 5 or more).
– 9,500 rounds to 10,000 (since 5 means round up).
4. Rounding Large Numbers (5-Digit and 6-Digit Numbers)
Rounding to the Nearest 10,000:
When rounding to the nearest 10,000, look at the thousands place.
– If the thousands place is 5 or more, round up.
– If the thousands place is less than 5, round down.
Examples:
– 34,567 rounds to 30,000(since 4 is less than 5).
– 65,432 rounds to 70,000 (since 5 or more means round up).
– 78,900 rounds to 80,000 (since 8 is more than 5).
Steps to Round Numbers:
1. Identify the place value you want to round to (ones, tens, hundreds, thousands, etc.).
2. Look at the digit to the immediate right of that place.
3. Apply the rounding rules:
– If 5 or more, round up.
– If less than 5, round down.
4. Change the digits after the rounding place to zero.
Roman numerals system
The Roman Numeral System is an ancient way of writing numbers using letters from the Roman alphabet. It’s different from our modern number system, but it’s still used today for special things like clocks, book chapters, and more!
Key Roman Numerals:
There are **seven** basic Roman numerals:
| Roman Numeral | Value |
|—————|———|
| I | 1 |
| V | 5 |
| X | 10 |
| L | 50 |
| C | 100 |
| D | 500 |
| M | 1,000 |
How to Write Roman Numerals:
You can combine these basic numerals to form other numbers. Roman numerals are written by adding or subtracting these values.
Addition Rule:
– You add the values when a smaller numeral is placed after a larger one.
Example:
– VI = 5 + 1 = 6
– XV = 10 + 5 = 15
Subtraction Rule:
– You subtract the values when a smaller numeral is placed before a larger one.
Example:
– IV = 5 – 1 = 4
– IX = 10 – 1 = 9
Important Roman Numeral Combinations:
– 1 to 10:
I, II, III, IV, V, VI, VII, VIII, IX, X
– 20, 30, 40, 50:
XX, XXX, XL, L
– 100, 200, 300, 400, 500, 1,000:
C, CC, CCC, CD, D, M
Rules to Remember:
1. Same numeral repeated up to three times: You can repeat a numeral up to three times in a row to add values.
– III = 1 + 1 + 1 = 3
– XXX = 10 + 10 + 10 = 30
2. Don’t use the same numeral more than three times in a row: After three repetitions, switch to subtraction.
– IIII is not allowed. Instead, write IV (5 – 1 = 4).
3. Subtraction is only used for certain numerals:
– I can be subtracted from V (5) and X (10) (e.g., IV = 4, IX = 9).
– X can be subtracted from L (50) and C (100) (e.g., XL = 40, XC = 90).
– C can be subtracted from D (500) and M(1,000) (e.g., CD = 400, CM = 900).
Examples of Roman Numerals:
– 23 = XXIII (20 + 3)
– 49 = XLIX (50 – 10 + 10 – 1)
– 76 = LXXVI (50 + 10 + 10 + 5 + 1)
– 142 = CXLII (100 + 50 – 10 + 1 + 1)
– 399 = CCCXCIX (100 + 100 + 100 + 100 – 10 + 10 – 1)