Knowing Our Numbers
Counting and representing large numbers is a skill that has evolved over thousands of years. In the distant past, people could only handle small numbers, but through collective effort and struggle, they gradually learned to work with larger numbers and express them symbolically. This journey reflects the development of mathematics as a whole, driven by the growing needs of human society.
Numbers enable us to quantify and communicate information about the world around us. They allow us to count tangible objects and describe collections of objects using numerical symbols or names. As our understanding of numbers expanded, so did our ability to comprehend and manipulate them, leading to further advancements in mathematics.
Comparing Numbers
- The number with the most number of digits is the largest number by magnitude, and the number with the least number of digits is the smallest number.Ex: Consider numbers: 252, 23, 4, 645, 5015. The largest number is 5015, and the smallest number is 4.
- The number with the highest leftmost digit is the largest number. If this digit also happens to be the same, we look at the next leftmost digit and so on.Ex: 840, 847, 650, 180, 235. The largest number is 847 and the smallest number is 180.
How many numbers can you make?
- If a certain number of digits are given, we can make different numbers having the same number of digits by interchanging the positions of digits.
- Example: Consider 4 digits: 5, 1, 9, 0. Using these four digits,
(a) Largest number possible = 9510
(b) Smallest number possible = 1059 (Since 4 digit number cannot have 0 as the leftmost number, the number then will become a 3-digit number)
Ascending Order and Descending Order
a) Ascending Order -Arranging numbers from smallest to greatest.
b) Descending Order – Arranging numbers from greatest to smallest.
Example- Consider group of numbers 387, 908, 4502, 4605, 7820, 345
. Ascending order- 345, 387, 908, 4502, 4605
.Descending order- 4605, 4502, 908, 387, 345
Shifting Digits
Take any three-digit number with different digits and interchange the digit at the hundreds place and the digit at the ones place. We find that the new number is greater than the former number, if the digit at the ones place is greater than the digit at the hundreds place and the new number is smaller than the former number, if in the former number the digit at the ones place is smaller than the digit at the hundreds place.
Example- consider number 4571
After interchanging digit of thousands place and tens place the new number formed is 1574.
Revisiting place value
The place value1 of a digit at ones place is the same as the digit. The place value of a digit at tens place is obtained by multiplying the digit by 10. Similarly, the place value of a digit at hundreds place, thousands place, ten thousands place,… is obtained by multiplying the digit by 100, 1000, 10000, …, respectively.
Example- 2901
Here 1 is at ones place , 0 is at tens place, 9 is at hundreds place.
Indian system of numeration
Values of the places in the Indian system of numeration are Ones, Tens, Hundreds, Thousands, Ten thousands, Lakhs, Ten Lakhs, Crores and so on.
The following place value chart can be used to identify the digit in any place in the Indian system.
| Crores | Lakhs | Thousands | Ones | |||||
|---|---|---|---|---|---|---|---|---|
|
Tens |
Ones |
Tens |
Ones |
Tens |
Ones |
Hundreds |
Tens |
Ones |
Example-
3,96,152 = 3 × 1,00,000 + 9 × 10,000 + 6 × 1,000 + 1 × 100 + 5 × 10 +2 × 1
This number has 2 at one’s place, 5 at tens place, 1 at hundreds place, 6 at thousands place, 9 at ten thousands place and 3 at lakh place.
Number Name are also written based on the place value name. So Its number name is Three lakh ninty-six thousand one hundred fifty-two.
Use of commas in Indian system of numeration
Commas added to numbers help us read and write large numbers easily. As per Indian Numeration, Commas are used to mark thousands, lakhs and crores. The first comma comes after hundreds place (three digits from the right) and marks thousands. The second comma comes two digits later (five digits from the right). It comes after ten thousand place and marks lakh. The third comma comes after another two-digits (seven digits from the right). It comes after ten lakh place and marks crore.
Example-
a) 54, 890, 345
b) 45,435
c) 23, 60, 450
International system of numeration
Values of the places in the International system of numeration are Ones, Tens, Hundreds, Thousands, Ten thousands, Hundred thousands, Millions, Ten millions and so on.
1 million = 1000 thousands,
1 billion = 1000 millions
| Billions | Millions | Thousands | Ones | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
|
Hundreds |
Tens |
Ones |
Hundreds |
Tens |
Ones |
Hundreds |
Tens |
Ones |
Hundreds |
Tens |
Ones |
Use of commas in International system of numeration
As per International Numeration, Commas are used to mark thousands and millions. It comes after every three digits from the right. The first comma marks thousands and the next comma marks millions. For example, the number 10,101,592 is read in the International System as tem million one hundred one thousand five hundred ninety-two. In the Indian System, it is 1 crore one lakh one thousand five hundred ninety-two.
Estimation
There are a number of situations in which we do not need the exact quantity but need only an estimate of this quantity. Estimation means approximating a quantity to the desired accuracy.
Estimating to the nearest tens by rounding off
The estimation is done by rounding off the numbers to the nearest tens. Thus, 17 is estimated as 20 to the nearest tens; 12 is estimated as 10 to the nearest tens.
Example – Consider number 54, 678 rounding to the nearest tens is 54,680.
Estimating to the nearest hundreds by rounding off
Numbers 1 to 49 are closer to 0 than to 100. So they are rounded off to 0. Numbers 51 to 99 are closer to 100 than to 0, and so are rounded off to 100. Number 50 is equidistant from 0 and 100 both. It is customary to round it off as 100.
Example – Consider number 54, 678 rounding to the nearest hundreds is 54,700.
Estimating to the nearest thousands by rounding off
Numbers 1 to 499 are nearer to 0 than 1000, so these numbers are rounded off as 0. The numbers 501 to 999 are nearer to 1000 than 0, so they are rounded off as 1000. Number 500 is customarily rounded off as 1000.
Example – Consider number 54, 678 rounding to the nearest thousands is 55,000.
Roman Numerals
- Digits 09 in Roman are represented as I, II, III, IV, V, VI, VII, VIII, IX, X
- Some other Roman numbers are : I = 1, V = 5 , X = 10 , L = 50 , C = 100 , D = 500 , M = 1000
Rules of the system
1) In Roman numerals a symbol is not repeated more than three times, but the symbols V, L and D are never repeated.
2) Roman numerals are read from left to right and the letters of Roman numerals are arranged from the largest to the smallest.
3) If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets added to the value of greater symbol.
VI = 5 + 1 = 6
4) If a symbol of smaller value is written to the left of a symbol of greater value, its then value is subtracted from the value of the greater symbol.
IV = 5 – 1 = 4
5)The symbol I can be subtracted from V and X only.
The symbol X can be subtracted from L, M and C only.
Whole Numbers
Whole numbers are a set of numbers that include all the positive integers (numbers greater than zero) along with zero. They do not include any fractions or decimals.
Example – 0, 1, 2, 3, 4, 5…….
Predecessor of a Number
The predecessor of a number is simply the number that comes just before a given number in the sequence of natural numbers.
For example, let’s consider the number 7:
- The predecessor of 7 is the number that comes immediately before it in the sequence of natural numbers, which is 6.
- If you are asked to find the predecessor of a number like 7, you simply subtract 1 from that number.
- So, the predecessor of 7 is 7−1=6.
Similarly:
- The predecessor of 15 is 15−1=14.
- The predecessor of 3 is 3−1=2.
Successor of a Number
The successor of a number is simply the number that comes just after a given number in the sequence of natural numbers.
For example, let’s consider the number 5:
- The successor of 5 is the number that comes immediately after it in the sequence of natural numbers, which is 6.
- If you are asked to find the successor of a number like 5, you simply add 1 to that number.
- So, the successor of 5 is 5+1=6.
Similarly:
- The successor of 12 is 12+1=13.
- The successor of 8 is 8+1=9.
The Number Line
The number line is a visual representation of numbers arranged in order from left to right, with each point on the line corresponding to a specific number.
Structure of the Number Line:
- The number line is a straight line that extends infinitely in both directions.
- It is marked with points at regular intervals, each representing a specific number.
Placement of Numbers:
- The number line typically starts with 0 at the center or leftmost side.
- Numbers increase as you move to the right and decrease as you move to the left.
Whole Numbers and Integers:
- On the number line, whole numbers (positive integers like 1, 2, 3…) are located to the right of 0.
- Negative integers (like -1, -2, -3…) are located to the left of 0.
- The number 0 is the point where the positive and negative numbers meet.
Representation of Numbers:
- Each point on the number line corresponds to a specific number.
- For example, the point directly to the right of 0 represents the number 1, the next point represents 2, and so on.
- Similarly, the points to the left of 0 represent negative numbers, such as -1, -2, and so forth.
Locating Numbers:
- To locate a number on the number line, find its position relative to 0.
- Move right for positive numbers and left for negative numbers until you reach the desired point corresponding to the number.
Understanding Distance:
- The distance between two points on the number line represents the difference between the corresponding numbers.
- For example, the distance between 3 and 7 on the number line is 4 units.
Addition on Number line
1. Start at the First Number:
– Choose the first number you want to add (the addend) and locate it on the number line.
– For example, let’s say we want to add 3 to another number.
2. Move to the Right:
– Starting from the first number, count units to the right on the number line.
– The number of units you count represents the second number (the number you are adding).
3. Find the Sum:
– Continue counting units to the right until you reach the second number.
– The point where you land after counting represents the sum of the two numbers.
Example:
Let’s add 3 and 5 using a number line:
– Start by locating the first number, which is 3, on the number line.
– From the number 3, count 5 units to the right (since we want to add 5).
– Each unit you count represents one number.
– When you reach the 5th unit from 3, you land on the number 8.
Therefore, 3 + 5 = 8.
Visualization on Number Line:
1. Locate 3 on the number line.
2. Count 5 units to the right.
3. You will reach 8, which is the sum of 3 and 5.
Explanation:
– Adding on a number line helps visualize the concept of combining two quantities.
– The starting point (first number) is fixed, and you move a certain number of units (second number) to find the sum.
Subtraction on Number line
Subtraction on a number line is a way to visually understand the concept of taking away or finding the difference between two numbers.
1. Start at the First Number:
– Choose the first number (the minuend) and locate it on the number line.
2. Move to the Left:
– Determine the second number (the subtrahend) that you want to subtract.
– From the first number, count units to the left on the number line corresponding to the value of the subtrahend.
3. Find the Result:
– The point where you land after counting to the left represents the result of the subtraction.
Example:
Let’s subtract 4 from 9 using a number line:
– Start by locating the first number, which is 9, on the number line.
– From the number 9, count 4 units to the left (since we want to subtract 4).
– Each unit you count represents one number.
– When you reach the 4th unit to the left of 9, you land on the number 5.
Therefore, 9 – 4 = 5.
Visualization on Number Line:
1. Locate 9 on the number line.
2. Count 4 units to the left.
3. You will reach 5, which is the result of 9 – 4.
Explanation:
– Subtraction on a number line helps visualize the concept of taking away or finding the difference between two quantities.
– Starting from the first number, you move backward (to the left) a certain number of units (corresponding to the subtrahend) to find the result.
Multiplication on Number line
Multiplication on a number line is a way of representing repeated addition or groups of a number. It helps visualize how multiplication works and how the product relates to the placement of numbers on a number line.
Understanding Multiplication on a Number Line:
1. Repetition of Addition: Multiplication can be thought of as repeated addition. For example, 3 times 4 means adding 3 four times 3 + 3 + 3 + 3 .
2. Number Line Representation: On a number line, we can start from zero and move along a specific distance (or jump) repeatedly.
Example:
Let’s consider 4 times 3. This means we want to add 4 three times:
– Start at 0: Begin at zero on the number line.
– First Jump: Jump 4 units to the right (positive direction) to represent 4.
– Second Jump: From where you landed (at 4), jump another 4 units to the right to represent the second 4.
– Third Jump:Again, jump 4 units to the right from the last position.
To find 4 times 3, you’re effectively making three jumps of 4 units each:
– First jump: 0 + 4 = 4
– Second jump: 4 + 4 = 8
– Third jump: 8 + 4 = 12
So, 4 times 3 = 12.
Representation on the Number Line:
On the number line:
– Start at 0.
– First jump to 4.
– Second jump to 8.
– Third jump to 12.
You can see that 4 times 3 is represented by the endpoint of the third jump, which lands at 12 on the number line.
Properties of Whole Numbers
1. Closure Property of Whole Numbers
The closure property of whole numbers means that when you perform certain operations (addition or multiplication) with whole numbers, the result is always a whole number. In other words, if you add or multiply any two whole numbers, the result will also be a whole number.
Example 1: Closure under Addition
Let’s take two whole numbers, 3 and 5.
3 + 5 = 8
Here, both 3 and 5 are whole numbers, and their sum (8) is also a whole number. This illustrates the closure property of addition for whole numbers.
Example 2: Closure under Multiplication
Now consider another pair of whole numbers, 4 and 2.
4 times 2 = 8
In this case, multiplying 4 and 2 results in 8, which is also a whole number. Therefore, multiplication of two whole numbers yields another whole number, demonstrating the closure property of multiplication for whole numbers.
In summary, the closure property of whole numbers ensures that the set of whole numbers is closed under both addition and multiplication, meaning that the result of adding or multiplying any two whole numbers will always be another whole number.
If any two whole numbers are subtracted, one may or may not get a whole number as a result. Whole numbers are not closed under subtraction.
Example
0 − 5 = Not a whole number
7 − 9 = Not a whole number
9 − 3 = 6
In the first two instances, the results are not whole numbers. Only in the third case, the result of subtraction is a whole number.
Similarly, if a whole number is divided by another whole number, one may or may not get a whole number as a result. So, whole numbers are not closed under division.
Example
0/6 = 0
2/7 , 2/7 is not a whole number.
6/2 = 3
In the first and third case, the division of any two whole numbers results in a whole number. But in the second case, the result is a fraction and not a whole number.
2.Commutative Property of Whole Numbers
This property applies specifically to addition and multiplication of whole numbers. Let’s break it down:
1. Commutative Property of Addition:
This property states that when you add two whole numbers, the order in which you add them does not change the result. In mathematical terms, for any whole numbers a and b,
a + b = b + a
For example, 2 + 3 = 3 + 2. Both calculations result in 5, showing that addition is commutative.
2. Commutative Property of Multiplication:
This property states that when you multiply two whole numbers, the order of multiplication does not affect the product. In mathematical terms, for any whole numbers a and b,
a times b = b times a
For example, 4 times 5 = 5 times 4. Both calculations result in 20, demonstrating that multiplication is commutative.
Addition:
a) 3 + 4 = 4 + 3
b) 5 + 2 = 2 + 5
Multiplication:
a) 2 times 6 = 6 times 2
b) 7 times 3 = 3 times 7
Addition and multiplication of whole numbers are commutative, but subtraction and division of whole numbers are not commutative. For example,
3 − 8 = Not a whole number
8 − 3 = 5
So subtraction is not commutative.
Similarly, division is not commutative either. For example,
3/6 = 1/2
6/3 = 2
3. Associative Property of Whole Numbers
When three or more whole numbers are added or multiplied, the results remain the same regardless of the grouping of the numbers. This is called associative property of whole numbers.
1. Associative Property of Addition:
This property states that when you are adding three or more whole numbers together, the grouping of the numbers does not affect the sum. In mathematical terms, for any whole numbers a, b, and c,
(a + b) + c = a + (b + c)
This means that whether you first add a and b, and then add c, or first add b and c, and then add a, the final sum remains the same.
For example:
(2 + 3) + 4 = 2 + (3 + 4) = 9
2. Associative Property of Multiplication:
This property states that when you are multiplying three or more whole numbers together, the grouping of the numbers does not affect the product. In mathematical terms, for any whole numbers a, b, and c,
(a times b) times c = a times (b times c)
This means that whether you first multiply a and b, and then multiply by c, or first multiply b and c, and then multiply by a, the final product remains the same.
For example:
(2 times 3) times 4 = 2 times (3 times 4) = 24
If the same numbers are subtracted by grouping them differently, we will get different results.
3 − (6− 2) = 3 − 4 = -1
(3 − 6) − 2 = -3 − 2 = -5
Similarly, division doesn’t follow associative property.
3 ÷ (6 ÷ 2) = 3 ÷ 3 = 1
(3 ÷ 6) ÷ 2 = 1/2 ÷ 2 = 1/4
Subtraction and division of whole numbers is not associative.
3. Distributive Property of Whole Numbers
The Distributive Property states that multiplication can be distributed over addition or subtraction. In other words, when you have a multiplication involving the sum or difference of two numbers, you can distribute the multiplication across each term inside the parentheses.
a) For Addition:
a times (b + c) = a times b + a times c
This means that when you multiply a number a by the sum of two numbers b and c , it is equivalent to multiplying a by b and then adding the product of a and c.
b) For Subtraction :
a times (b – c) = a times b – a times c
Similarly, when you multiply a number a by the difference of two numbers b and c, it is equivalent to multiplying a by b and then subtracting the product of a and c.
Examples
1. Using Addition:
3 times (4 + 2) = 3 times 4 + 3 times 2
3 times 6 = 12 + 6
18 = 18
2. Using Subtraction:
5 times (8 – 3) = 5 times 8 – 5 times 3
5 times 5 = 40 – 15
25 = 25
Identity Numbers
Consider the following:
2 + 0 = 2
5 + 0 = 5
Any number + 0 = Same number
This is called additive identity of whole numbers. So, 0 is the identity number for addition of whole numbers.
Identity numbers can be different for different operations like addition, subtraction, multiplication, and division.
Any number + 0 = Same number
Any number − 0 = Same number
Any number × 1 = Same number
Any number ÷ 1 = Same number
1 is the identity number for multiplication and division of whole numbers.
Patterns Using Whole Numbers
We shall try to arrange numbers in elementary shapes made up of dots. The
shapes we take are (1) a line (2) a rectangle (3) a square and (4) a triangle.
Every number should be arranged in one of these shapes. No other shape is allowed.
Straight Line
If two points are given, then the two can be joined together in the shape of a straight line
With one more point, we can still form a straight line.
We can form a straight line with any number of points, except 1.
Triangular numbers
Numbers that form a triangle are called triangular numbers. For example, 3, 6, 10, 15, and 21 are triangular numbers.
Square Numbers
We get a square number by multiplying a whole number with itself. 4, 9, 16… are the numbers that form square patterns.
If we add any two consecutive triangular numbers, we end up getting a square number. For example,
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
6 and 10 are consecutive triangular numbers, whereas 16 is a square number.
Rectangular Numbers
Whole numbers that can be arranged in a rectangular pattern are called rectangular numbers
A whole number is a rectangular number if it can be written as a product of two different whole numbers (one of the whole numbers should not be number 1).
Example: 12 is a rectangular number because it can be expressed as,
2 × 6 = 12
3 × 4 = 12
All square numbers are rectangular numbers, but the opposite
