Place Value and Integers 

Note: The position of a digit in a number determines its value.

Each digit in a number has a place which tells us the value of the digit. If the place of the digit in a number changes, so does its value.

 ‘Place Value’ is the position of a digit in a number which determines its value.

Place value is a way of understanding and organizing numbers. It helps us know the value of each digit in a number and where it belongs in relation to other digits. Let’s start by looking at a number like 345.

In the number 345, each digit holds a specific value based on its position. The rightmost digit is in the ones place, the next digit to the left is in the tens place, and the digit after that is in the hundreds place. So, let’s break it down:

• The digit 5 is in the ones place. That means it represents the value of 5 ones or simply 5.
• The digit 4 is in the tens place. That means it represents the value of 4 tens or 40.
• The digit 3 is in the hundreds place. That means it represents the value of 3 hundreds or 300.
• So, when we put these values together, we get the number 345, which is made up of 300 + 40 + 5.

Understanding place value helps us to read, write, and compare numbers. It allows us to know that a number like 345 is greater than 200 because the hundreds place has a value of 3, which is greater than 2.

Understanding place value helps us work with numbers more easily, whether it’s doing addition, subtraction, multiplication, or even more advanced math later on. It’s an essential concept to grasp when working with number.

Place value is a helpful tool for understanding numbers in many ways. First, it helps us count and keep track of large numbers by showing the value of each digit based on its position. Second, it allows us to read and write numbers correctly, knowing that each digit represents a different value depending on its place. Third, place value helps us compare numbers and determine which one is bigger or smaller by looking at the digits in each place. Fourth, it makes addition and subtraction easier because we can line up the digits and work with them one place at a time. Fifth, place value assists us in making good estimates by looking at the most significant digit. Lastly, it is essential for working with money and decimals, helping us understand the value of each digit after the decimal point. Understanding place value is like having a special code that helps us understand and work with numbers more easily!

The number 34 765 123 945 281 can be represented using a place value chart:

Adding a space between digits within a number helps in reading and understanding them better. Let us find out what the best spacing is?

Starting from the right, there must be a space after every three numbers. It helps in identifying the place value of individual digits.

Using the numbers we used above as an example, let us add spaces to the numbers.

Let’s practice. Identify the place value of the underlined digit in the following numbers.

Imagine a magical number line that stretches endlessly in both directions, like an infinite highway of numbers. On this enchanted road, you’ll encounter integers, which are positive and negative numbers. Integers are like characters with their own personalities. Positive numbers are the cheerful ones, representing things like the number of friends you have or the delicious slices of pizza you devour. Negative numbers, on the other hand, are the mysterious ones, symbolizing debts you owe or the depths of a deep, dark cave. As you journey along the number line, you’ll notice that zero acts as the meeting point for positive and negative integers, the balance between light and shadow. It’s a place where gains and losses cancel each other out. The number line becomes your guide, helping you understand the relationships between these integers. Moving to the right takes you into greater positive numbers, while moving to the left reveal greater negative numbers. You can add or subtract integers by hopping along the number line, and the distance you travel determines the result. The number line is a powerful tool that helps you explore the world of integers and understand their unique roles in the magical realm of numbers.

Integers are positive or negative whole numbers as shown on the number line below.

For addition and subtraction of positive and negative numbers,

we will use a number line.

If moving 3 places on the right on a number line is +3, then

 moving 5 places on the left would be −5.

Adding 2 positive numbers is simply addition.

For example: + 3 + 7 is like first moving 3 places right from 0 and then further moving 7 places right. So, we will land on 10. Therefore, + 3 + 7 = + 10

Similarly, adding 2 negative (−ve) numbers is also addition but the answer will carry a −ve sign. For example: − 5 − 2 means moving 5 places to the left of 0 and then again moving 2 places on the left.

We see that we finally land on −7.

Therefore, we conclude that: + x + y = + (x + y) and − x − y = − (x + y) OR − x + (− y) = − (x + y)

Now if we move 5 places towards the right (which is +5) and then we move 9 places towards the

left, we see that we land on −4.

We are on the left side because we are moving more towards the left as compared to the right.

Hence, we see that + 5 − 9 = − 4

Which means when +ve and −ve (both movements are involved) we find the difference of the two numbers and the sign of the number which has a greater value is carried in the answer. In the case above, the difference of 9 and 5 is 4 and since 9 > 5, we use the sign of 9 and the answer is −4.

Note: Subhan is learning horse riding. He trains for 4 hours per day on

 Saturday and Sunday.  

4 x 2 = 2 x 4 = 8

Let’s suppose that his parents know his total training hours and the number of

days he trains for in a week. How would they find the number of hours he trains for each day?

They can find that out by performing division:

8 ÷ 2 = 4 x 2 ÷ 2 = 4

The numbers used in the above equation are the same as that in the earlier equation. The only difference is in the operation applied (division in place of multiplication) and the places of 4 and 8. Instead of 4 being multiplied with 2, it is being divided by 8.

What division will Subhan’s parents need to carry out if they want to know the number of days trained, using the total number of hours trained and the hours trained for each day? It would simply be:

8 ÷ 4 = 2 x 4 ÷ 4 = 2

The reciprocal of a number is the result of dividing 1 by it. For example, the reciprocal of 8 is and the reciprocal of ⅛ is 1 ÷ ( ⅛ ) = 8.

The product of a number and its reciprocal is always 1. For example, 8 x 18 = 1 and 18 x 8 = 1.

Let us now learn how to multiply positive and negative numbers.

4 × 3 is read as 4 groups of 3’s. Similarly 4 × (− 3) means four groups of (− 3) which means (− 3) is added 4 times.