Open Number Line

A number line with no numbers marked on it is called an open or blank number line. It is a horizontal straight line with arrowheads at both ends.

As the numbers on a number line are never-ending, an open number line can plot the numbers according to our requirements during an arithmetic operation.

When placing numbers on an open number line, we first decide the range that best displays the number we need to represent. Here, the numbers are placed at equal intervals such that the distance between two consecutive numbers increases uniformly as we move to the right and decreases as we move to the left.

We can use different strategies while performing any operations on an open number line.

For example, we can plot numbers from 10 to 20, keeping an interval of 1 unit between each consecutive number.

We can also plot the same numbers with an interval of 2 units between two consecutive numbers.

Our approach depends on the number of ranges plotted on an open number line.

Finding Real Numbers

An open number line is often used to find a real number (integer, fraction, or decimal).

Let us now find a number, say 15, with the help of an open number line.

Keeping the range from 10 to 20 and moving 5 steps to the right of 10 on the number line, we find 15.

Now let us locate the fraction ${14\dfrac{3}{7}}$  on the above open number line. 

Since ${14\dfrac{3}{7}}$  lies between 14 and 15, the region from 14 to 15 is divided into 7 equal parts.

Next, moving 3 steps to the right starting from 14, we find ${14\dfrac{3}{7}}$.

Now let us place the decimal number 14.59 on an open number line.

As 14 < 14.59 < 15 and 14.5 < 14.59 < 14.6, we divide the area from 14.5 to 14.6 into 10 equal parts and move 9 steps to the right starting from 14.5, to get the number 14.59.

Adding Numbers

An open number line is used to add two or more numbers.

For example, 25 + 33 gives 58.

We can use different strategies to perform this addition using an open number line because 25 + 33 can be decomposed in various ways, like 20 + 30 + 8 or 25 + 30 +2 +1 or 25 + 5 + 20 +8, and so on.

Case 1: 25 + 33 = 20 + 30 + 8

Here we add the tens (20 + 30) and then the ones (5 +3).

1. The first number, 20, being positive, is marked to the right of 0 using an open number line.
2. A longer jump is taken to the right of 20 to count forward by 3 tens taken from the second number (i.e., 33), which takes us to 50.
3. Next, we get 8 by adding 5 +3 and thus take a shorter jump to count forward by 8 to the right of 50.

The result is 25 +33 = 20 + 30 + 8 = 58 on the number line.

Case 2: 25 + 33 = 25 + 30 +2 +1

Here 25 is added as a whole number, and then 3 tens are added from the second number (33). Next, 3 is added as 2 + 1.

1. As the first number 25 is positive, it is marked to the right of 0 using an open number line.
2. We count forward by 30 once to the right of 25 and get 55.
3. Then, breaking 3 into 2 + 1 and first moving 2 steps and then 1 step to the right of 55 gives us the final result.

Thus we find 25 + 33 = 25 + 30 +2 + 1 = 58 on the number line

Case 3: 25 + 33 = 25 + 5 + 20 + 8

Here we add 33 by decomposing it into 5 ones, 20, and 8 ones.

1. 25 is marked to the right of 0.
2. We count forward by 5 ones to the right of 25 and get 25 + 5 = 30.
3. A longer jump to count forward by 20 is taken to the right of 30 to get 50.
4. At last, the remaining 8 are added by moving to the right of 50.

Thus the result 25 + 33 = 25 + 5 + 20 + 8 = 58 is obtained on the number line.

Subtracting Numbers

Similar to addition, we can also use an open number line to subtract two or more numbers.

For example, 78 – 37 gives 41, where 78 is the minuend, 37 is the subtrahend, and 41 is the difference or the result.

We use different strategies to perform this subtraction similar to addition and observe that the result is the same for each case.

Case 1: 78 – 37 = 78 – 8 – 20 – 9

Here the subtrahend 37 is decomposed into 8 + 20 + 9 and subtracted from the minuend 78 by following the steps below.

1. 78 being positive, is marked on an open number line to the right side of 0.
2. First, 8 ones are subtracted by moving 8 steps to the left of 78 to get a ten out of it (70)
3. Then, a long jump is taken to count backward by 20 to the left of 70, and we get another ten, i.e., 50.
4. Finally, 9 ones are subtracted by moving 9 steps to the left of 50.

So we find the result 78 – 37 = 78 – 8 – 20 – 9 = 41 on the number line.

Case 2: 78 – 37 = 78 – 30 – 7

In this case, we break the subtrahend 37 as 30 + 7 and get the result by subtracting the tens (30) and the ones (7) simultaneously from the minuend 78.

1. 78 is plotted to the right of 0.
2. First, we take a long jump to the left of 78 to count backward by 30 once and get 48.
3. 7 ones are subtracted by moving 7 steps to the left of 48.

Thus the result 78 – 37 = 78 – 30 – 7 = 41 is obtained on the number line.

Multiplying Numbers

As multiplication is repeated addition, an open number line helps to find the result after repeating the addition for the required number of times.

For example, 5 × 9 is the same as adding  9 five times.

So to find 5 × 9 on a number line, we take 5 jumps of equal size, i.e., 9 to the right starting from 0.

Thus 5 × 9 = 45 is obtained on the number line.

Dividing Numbers

We can also perform division on an open number line.

For example, to find 350 ÷ 50, we first draw an open number line with the first few multiples of 50 and mark 350 there.

Since division is repeated subtraction, the result is obtained after subtracting 50 from 350 several times until we reach 0.

We then move backward by 50 steps each time to get 0.

Thus the result is 7 as the process is repeated seven times.

Rounding and Comparing Numbers

A number can be rounded up or down to its closest given number using an open number line.

To round 5568 to its closest ten, we write the numbers from 5560 to 5570 on an open number line and mark the mid-point 5565.

As 5568 > 5565 lies to the right of 5565, 5568 is closer to 5570.

Thus 5568 is rounded up to 5570.

Finding Elapsed Time

An open number line also helps us find starting and ending times and the time elapsed between two events.

For example, let us find the time between 2:14 AM and 5:02 AM.

Here the minuend is 5:02, and the result of the subtraction is 2:14. We need to calculate the subtrahend.

1. First, 5:02 is marked to the right of 0.
2. 2 minutes are subtracted by moving to the left of 5:02 to get exactly 5:00.
3. Then, 45 more minutes are subtracted by moving to the left of 5:00 to get 4:15
4. We further move 1 step to the left of 4:15 to get 4:14 
5. Finally, moving 2 hours, i.e., 120 minutes to the left of 4:14, takes us to the result 2:14

So the elapsed time between 2:14 AM and 5:02 AM is given by adding up all the minutes subtracted in the above steps.

Thus the total time elapsed is 120 + 1 + 45 + 2 = 168 minutes = 2 hours and 48 minutes.