**What is the relative value of a number?**

He **relative value of a number** or digit of the decimal system depends on the position it occupies when it is part of a number. Therefore, it is said to be a place value. A very simple example: the **relative value of 1** at number 123, **it will be 100**since the 1 occupies the hundreds place.

**Another example**: he number 58 is made up of digits 5 and 8. Examining this number from right to left, we have that the relative value of 8 is 8because it is in the units position and the relative value of 5 is 50, for occupying the tens place. The number is read «fifty-eight.»

Instead, the same digits have different relative values in the number 85, since they have exchanged positions. Always starting from right to left, the relative value of 5 in this case is 5he relative value of 8 is 80 and the number is read «eighty-five.»

**How to find the relative value of a number?**

The general procedure for finding the relative value of each digit is as follows:

The first digit from right to left is in the ones place and its value is multiplied by 1.

The next digit corresponds to the tens place and is multiplied by 10.

The next position corresponds to the hundreds place and the value of the digit is multiplied by 100.

The next position is the thousand, so the digit is multiplied by 1000.

And so on for larger numbers, multiplying the digit by the corresponding power of 10: 100,000, 1,000,000, and beyond.

For example, the number 321 can be written as 3*100 + 2*10 + 1*1, or equivalently 300+20+1. In the example above, you can quickly see that the relative value of 3 is 300, of 2 is 20, and of 1 is 1.

**Examples of relative values**

**number 727**

To determine the relative value of a digit, one must be guided by the following basic principle of the written numbering of the decimal system:

*Any digit to the left of another represents a unit 10 times greater, and conversely: any digit to the right of another represents a unit 10 times less. *

For example, the number 727, which is read «seven hundred twenty-seven», consists of the digits 2 and 7, with the 7 repeated, but occupying different positions.

Reading 727 from right to left, it is observed that the 7 on the right occupies the unit position, therefore it is multiplied by 1:

7 x 1 = 7

And its relative value is 7.

The digit 2 in the middle occupies the tens place and to find its relative value it is multiplied by 10:

2 x 10 = 20

Finally, the 7 to the extreme left has the hundreds place. Then you have to multiply it by 100 and its relative value is:

7 x 100 = 700

Note that only when the digit occupies the unit position is that its absolute value and its relative value are equal. Therefore, if the relative value of the number is VR and its absolute value is VA, the general formula for finding the relative value is:

VR = absolute value VA × value of your position

A number can be written as the sum of the relative values of its digits, this is known as *expanded notation*. Continuing with the example of the number 727, we have:

727 = 700 + 20 + 7

And if you prefer to use the powers of 10, the number 727 is also expressed in an equivalent way as:

727 = 7∙102 + 2∙101 + 7∙100

where the exponents of the power in base 10 represent the position of each digit and are called *indices*. Another example is illustrated in the following figure.

**number 63**

Starting from left to right, the 3 is in the ones place, therefore:

Relative value of 3: 3 x 1 = 3

As for the 6, it is in the tens place, so:

Relative value of 6: 6 x 10 = 60

**number 603**

This figure is different from the previous one, because although the relative value of 0 is 0, the other digits have different relative values. Starting from right to left as usual:

Relative value of 3: 3 x 1 = 3

Relative value of 0: 0 x 10 = 0

Relative value of 6: 6 x 100 = 600

**number 630**

In this case, 0 is in the units position:

Relative value of 0: 0 x 1 = 0

Relative value of 3: 3 x 10 = 30

Relative value of 6: 6 x 100 = 600

**solved exercises**

**Exercise 1**

Indicate the relative value of the underlined numbers:

to) 1209

b) 2782

c) 376

d) 3045

d) 273

**Solution**

a) The digit 1 in 1209 occupies the thousand or thousands position. Therefore, its relative value is 1000.

VR(1) = 1 x 1000 = 1000

b) The 2 occupies the units position in 2782, therefore its relative value is 2.

c) In 376 the 7 is in the tens place and:

VR(7) = 7 x 10 = 70.

d) In 3045 the 4 is also in the tens place:

VR(4) = 4 x 10 = 40.

e) For 273, the 3 is in the units place and its relative value coincides with the figure of the digit, that is:

VR(3) = 3 x 1 = 3

**Exercise 2**

Write the smallest number of 5 digits, without any being repeated and fulfilling the following conditions:

a) All digits are different

b) It has a 7 in the thousands place

c) The 8 is in the units position.

**Solution to**

The smallest number of 5 digits, with all of them different, must begin with 1, since although 0 is less, as the first digit to the left it does not count, therefore the number sought is:

10234

**solution b**

The thousand place for 7 corresponds to 7000, but since you want the smallest possible number containing 5 digits, the number has to start with 1, followed by 7 and then 023 in the remaining places, since no digit should be repeated .

Therefore the number is:

17023

**solution c**

Since the 8 is required to be in the ones place, it must be to the far right. Being the smallest number possible, without any of its 5 digits being repeated, the number sought is:

10238

**Exercise 3**

Calculate the absolute and relative value (of each digit) of the number 579.

**Solution**

One has that 579 is equal to 5×100+7×10+9×1, or equivalently, it is equal to 500+70+9. Therefore the relative value of 5 is 500, the relative value of 7 is 70, and the relative value of 9 is 9.

On the other hand, the absolute value of 579 is equal to 579.

**Exercise 4**

Given the number 9,648,736, what is the relative value of the 9 and the first 6 (from left to right)? What is the absolute value of the given number?

**Solution**

When rewriting the number 9,648,736, it is obtained that this is equivalent to

9×1,000,000 + 6×100,000 + 4×10,000 + 8×1,000 + 7×100 + 3×10 + 6×1

or it can be written as

9,000,000 + 600,000 + 40,000 + 8,000 + 700 + 30 + 6.

So the relative value of 9 is 9,000,000 and the relative value of the first 6 is 600,000.

On the other hand, the absolute value of the given number is 9,648,736.