NCERT Solutions for Class 6 Maths Chapter 6 Exercise 6.1

Class VI Mathematics NCERT (https://ncert.nic.in/) Exercise 6.1 of chapter 6 Integers in Hindi and English Medium free to use online or download in PDF file format for session 2024-25. All the solutions and book for class 6 mathematics exercise 6.1 are updated for academic session 2024-25. NCERT Class 6 Maths Exercise 6.1 give a brief idea about plotting numbers on number line. All the questions are solved step by step with suitable diagrams.

We know that if we subtract a whole number from another whole number, the difference may not always be a whole number. In fact, when a smaller number is subtracted from a larger number, we get a whole number. For example, 5 – 4 = 1, 25 – 13 = 12, 40 – 26 = 14 etc. are whole numbers. But, we have no whole numbers to represent 6 – 8, 16 – 40, 15 – 20 etc. So, there is a need to extend our number system so that we have numbers to represent such differences.

Corresponding to natural numbers1, 2, 3, 4, 5, 6, …etc.; we introduce new numbers –1, –2, –3, –4, –5, –6, …etc., called minus one, minus two, minus three, minus four, minus five etc. respectively such that 1 + (–1) = 0, 2 + (–2) = 0, 3 + (–3) = 0 and so on. –1 is also called negative of 1 and –1 and 1 are called the opposites of each other.

Combining these new numbers with whole numbers, we obtain a new collection of numbers which is written as …, –3, –2, –1, 0, 1, 2, 3, … These numbers are called integers.
The numbers 1, 2, 3, 4, 5, …, i.e., natural numbers are called positive integers. The numbers –1, –2, –3, –4, –5, …are called negative integers. The number zero is an integer, which is neither positive nor negative.

Use of integers is quite common in our day-to-day life. For example, positive and negative integers are used to represent concepts such as height above and below sea level; profit and loss; rise and fall in prices etc. In mathematical terms, we represent above sea level, profit, rise in price etc., by positive integers and height below sea level, loss, fall in price etc., by negative integers.

For example:
if the height 5 km above sea level is represented by +5, then the height below sea level will be represented by –5. Similarly, if ‘a profit of Rs. 50 in a transaction’ is represented by +50, then ‘a loss of Rs 50 in a transaction’ will be represented by –50.

In order to represent integers on a number line, we draw a line and mark a point O almost in the middle of it as shown in Fig., Now, we let off equal distances on right hand side as well as on the left hand side of O. On the right side, we label to the point of divisions as 1, 2, 3, 4, 5 etc. and the point O as 0. Since, negative integers are opposite of positive integers, namely natural numbers, therefore, we represent negative integers –1, –2, –3, –4, –5 etc.in the opposite direction, i.e., on the left side on the number line.

Clearly, 1 and –1 are at equal distances from 0 but in the opposite directions. Similarly, 2 and –2 are at equal distances from 0 but in opposite directions and so on.

Take a coin. Put it on a table with face ‘tail’ up. Turn the coin so that the face ‘head’ is up. Now turn the coin again. What do you find? This brings the face ‘tail’ up, i.e., the coin is in its original position. Therefore, negative of a negative integer is a positive integer, i.e., – (–a) = a for any integer a.

We have learnt that a whole number on the number line is greater than every whole number to its left. Thus, 7 > 4.
If we, now, follow the same criterion for the number line extended for integers, then an integer is greater than every integer to its left.
Thus, 3 > 2, since 3 is to the right of 2;
1 > 0, since 1 is to the right of 0;
0 > –2, since 0 is to the right of –2;
–4 > –5, since –4 is to the right of –5 on the number line.

(i) Every positive integer is greater than every negative integer.
(ii) Every positive integer is greater than zero.
(iii) Every negative integer is less than zero.
(iv) If a and b are any two integers such that a > b, then –a < –b. (v) If a and b are any two integers such that a < b, then –a > –b.

For example, 6 > 4 and –6 < –4. Also 6 < 9 and –6 > –9.