Place value and powers of ten

Why is the $7$ in $700$ worth so much more than the $7$ in $7$? Because of place value. Each column tells the digit what to multiply itself by.

Idea

In any decimal numeral, the value of a digit is the digit times the value of its column. From right to left, the columns are: $1, 10, 100, 1000, 10000, 100000, \dots$ or, using powers of $10$, $10^0, 10^1, 10^2, 10^3, 10^4, 10^5, \dots$

The exponent simply tells how many zeros come after the $1$. So $10^5 = 100000 = 1$ lakh, and $10^7 = 1$ crore.

Any large number can be written as a sum of digit $\times$ place value. This is called expanded form. For example, $4{,}07{,}523 = 4 \times 10^5 + 0 \times 10^4 + 7 \times 10^3 + 5 \times 10^2 + 2 \times 10 + 3.$

The expanded form is useful because it lets you see at a glance which column is causing a difference between two numbers. It also explains why we sometimes write large numbers as $3.5 \times 10^9$ instead of $3500000000$ , fewer zeros, less chance to miscount.

You will use place value every time you add, subtract, compare or round large numbers. Master it once and the rest of the chapter feels easy.

Worked examples

Example 1. Write $6{,}48{,}029$ in expanded form.

$6{,}48{,}029 = 6 \times 10^5 + 4 \times 10^4 + 8 \times 10^3 + 0 \times 10^2 + 2 \times 10 + 9.$

Example 2. What is the value of the digit $5$ in $35{,}24{,}716$?

The $5$ sits in the ten-lakh column ($10^6$), so its value is $5 \times 10^6 = 50{,}00{,}000$ (fifty lakh).

Example 3. Find the number whose expanded form is $7 \times 10^7 + 3 \times 10^5 + 6 \times 10^3 + 2$.

Filling in the missing columns with zero gives $70{,}306{,}002$, written in Indian style as $7{,}03{,}06{,}002$ , seven crore, three lakh, six thousand and two.

Example 4. Without doing full multiplication, find $1$ crore $\div 1$ lakh.

$1$ crore $= 10^7$ and $1$ lakh $= 10^5$. So $\frac{10^7}{10^5} = 10^{7-5} = 10^2 = 100.$ There are $100$ lakhs in $1$ crore.

Try it yourself

1. Write $58{,}07{,}403$ in expanded form.
2. What is the place value of the underlined digit in $9\underline{4}{,}26{,}108$?
3. Write $3 \times 10^6 + 2 \times 10^4 + 5 \times 10^2$ as a single numeral.
4. How many lakhs make a billion?
5. Express $2{,}50{,}00{,}000$ as a power of $10$ times a small number.
6. Find the difference in place value between the two $5$'s in $55{,}55{,}555$ (leftmost and rightmost).
7. If a number has the digit $9$ in the crore place and zeros everywhere else, write it down and read it.
8. Which is bigger: $10^8$ or $5 \times 10^7$? By how much?

Activity

Take any $8$-digit number (e.g., your phone-bill amount or a population). Write its expanded form on chart paper. Now scramble the digits and ask a friend to figure out the new value of each digit. Discuss what changes when you move a $5$ from the ones to the ten-lakh column.