Tecchilaw - 5. Number line

Tecchilaw

By studying this lesson, you will be able to,

● identify the number line,

● identify negative numbers,

● identify integers,

● represent integers on a number line and

● compare integers.

5.1 Marking whole numbers on a number line

Some of the instruments that we use when we perform various tasks are marked with numbers. A ruler which has been calibrated as mentioned is shown below.

Observe whether there are similarities between the ruler depicted above and the ruler in your instrument box.

Several properties which are similar that were obtained through such an observation are given below.

(i) The measuring edge of a ruler is made straight.

(ii) The whole numbers 0, 1, 2, 3, … have been marked on it with equal gaps between the numbers, starting from 0 and gradually increasing in value.

Such calibrations can also be observed in a spring balance used to measure weights and a measuring cylinder used to measure liquid amounts.

● A line such as this, which is used to represent numbers is called a number line.

● An arrowhead is drawn at the right end of the number line.

● The values of the numbers on a number line increase gradually towards the right.

● The difference between two numbers which are next to each other on the above number line is 1. Two such whole numbers, where the difference between them is 1, are called consecutive whole numbers.

● The quantitative information of certain things can be represented on a number line.

● A number is marked on a number line by placing a dot as shown below.

The numbers 2 and 4 have been marked on the above number line.

Let us consider an instance of using the number line to represent quantitative information.

A grade 6 student knows that the lengths of his eraser, pencil and pen are 4 cm, 10 cm and 14 cm respectively. When these three numerical values are marked on a number line, it is as follows.

Accordingly , it is clear that the following statements are true.

(i) The length of the pen is greater than the length of the pencil.

(ii) The length of the eraser is less than the length of the pen.

(iii) The length of the pencil is 6 units greater than the length of the eraser.

Exercise 5.1

(1) Copy the following number line. Mark the numbers 1, 2 and 5 on it.

(2) Write down the numbers that have been marked on the number line given below.

(3) Write down two special features of a number line.

(4) Mark the numbers 4, 7 and 2 on a number line.

(5) Nimal is 8 years old. His younger sister is 5 years old. Mark these values on a number line.

(6) Write down the numbers that have been marked on the number line given below.

(7) Geetha’s and Nimal’s ages in years have been indicated on the number line given below.

(i) Who is older from Geetha and Nimal?

(ii) What is Geetha’s age now?

(iii) How old will Nimal be when Geetha is 10 years old?

5.2 Negative numbers

The distance from 0 to 1 and from 0 to -1 is the same. Similarly, the number corresponding to the point which is two gaps to the left of 0 is called negative two. It is denoted by -2. Here, the distance from 0 to 2 and from 0 to -2 is the same.

In the same manner, moving further towards the left from 0, mark the rest of the points as -3, - 4 and -5

The whole numbers to the right of zero on this number line are called positive integers. The positive integers are 1, 2, 3, 4, … and so on. The three dots to the right of the numbers indicate that there are more numbers in the given manner.

The numbers to the left of zero on the number line are negative numbers.The negative whole numbers to the left of zero are called negative integers. The negative integers are -1, -2, -3, … and so on. These negative numbers are also indicated in the following manner; …, -3, -2, -1.

The number zero is neither positive nor negative.

The positive integers, the negative integers and zero, all taken together are called the integers. The integers are … , -3, -2, -1, 0, 1, 2, 3, …

There are many instances where negative numbers are used. One such case is described below.

There are many places in the world, where the temperature drops below zero degrees Celsius. The maximum and minimum temperatures recorded in five main cities around the world on a particular day are given in the following table.

A certain standard temperature is taken as 0 o C. In this table, certain temperatures such as -2 o C, -5 o C and -10 o C have the negative sign written in front of the number. This is used to indicate that those temperatures are less than the above standard temperature.

Similarly, in thermometers which are used to measure the temperature, the negative sign is written in front of the numbers to indicate temperatures which are below 0 o C.

Exercise 5.2

(1) Write down the numbers that have been marked on the number line given below.

(2) Write down the values represented by P, Q and R on the number line given below.

Value denoted by P =

Value denoted by Q =

Value denoted by R =

(3) Copy the number line given below and mark the numbers 4, 1 and -3 on it.

(4) On a number line from -5 to 5, mark the numbers 4, - 4 and -1 and name them as A, B and C respectively.

5.3 Comparison of integers

Let us consider the numbers five and two. We know that five is greater than two. We can write this as “five is greater than two”. The way this can be expressed concisely using symbols is given below.

5 > 2

Here the symbol “ > ” which expresses the meaning “greater than” has been placed between the numbers five and two. Similarly, the fact that 9 is greater than 4 can be expressed as 9 > 4. The fact that two is less than five can be expressed as follows.

2 < 5

The symbol “< ” expresses the meaning “less than”. Accordingly, 4 is less than 9 is symbolized as 4 < 9. When two integers are being compared, these symbols need to be used in the following manner.

The symbols “>” and “<” are called inequality signs.

The pointed side of these symbols is in the direction of the smaller number.

The above mentioned numbers have been marked on the number line given below.

Any number to the right of a particular number on the number line is greater (larger) than the said number. This property is applicable to the whole number line. Therefore, this rule can be applied to compare integers using the number line.

Let us consider which number from 0 and -2 is greater. Let us draw a number line and mark 0 and -2 on it.

On the number line, 0 is to the right of - 2. Therefore, 0 is greater than -2. This can be written as 0 > -2.

In the same manner, let us consider which number from -5 and -1 is greater.

Let us draw a number line and mark -5 and -1 on it.

On the number line, -1 is to the right of -5. Therefore, -1 is greater than -5. This can be written as -1 > -5

Exercise 5.3

(1) Given below are pairs of integers which have been compared using inequality signs. Write down how each inequality is described in words.

(2) For each of the following, write down whether the relationship is correct or incorrect.

(i) -5 > -8

(ii) -3 < 2

(iii) -7 > 0

(iv) -2 = 2

(v) 8 < -9

(vi) 6 < -4

5.4 Comparison of integers described further

The number line can easily be used to compare more than two numbers together.

For example,

let us take the integers 3, 0, -1 and -3 and compare them by using a number line.

On a number line, “the values of numbers increase gradually from left to right”. Therefore, if the above numbers are written in increasing order of their values, we get -3, -1, 0, 3. If we write numbers in increasing order of their values as above, we say that the numbers are written in ascending order.

The above numbers can also be written as 3, 0, -1, -3 with the values gradually decreasing. If we write several numbers in this manner in decreasing order of their values, we say that the numbers are written in descending order.

Exercise 5.4

(1) Write down the numbers marked on the number line, in ascending order

(2) Write down the following numbers in descending order with the aid of a number line.

-2, 2, 0, -4

(3) Write the following integers in ascending order with the aid of a number line.

0, -1, 2, -4, -2

(4) Consider the following number line on which the ages in years of three children have been represented.

(i) Write down the ages of the children in descending order.

(ii) Write down the names of the children in the order of decreasing age.

(iii) Who is the oldest child? Who is the youngest child?

(5) The average temperature in degrees Celsius of several cities around the world on a certain day, has been marked on the number line given below.

According to the number line,

(i) which city recorded the lowest temperature?

(ii) which city recorded the highest temperature?

(iii) on that day, how many units less was the average temperature in Peking than the average temperature in New Delhi?

(iv) when the difference between the temperatures of New Delhi and Peking and of New Delhi and Melbourne are considered, which is greater?

5.5 Finding integers between two non-consecutive integers

Sitha is 10 years old. Her brother, Madhava is 6 years old. Sriya who lives in a nearby house comes often to play with the two of them. Her age in years is between 6 and 10.

The integers between 6 and 10 are only 7, 8, and 9. Therefore, Sriya's age in years can be considered to be either 7, 8 or 9. This can easily be found using a number line too.