1.1.2 · D5Arithmetic & Number Systems
Question bank — Place value system — units, tens, hundreds, thousands, lakhs, crores
Before we start, the two words that cause 90% of the confusion — pin them down:
True or false — justify
Every column is worth ten times the column on its right.
True. We count in tens, so when a column fills past it resets to and carries one leftward — that "-for-" swap is exactly what makes each left column bigger.
The place value of a digit can never be smaller than its face value.
True for whole numbers. The place value is face value a power of ten (), and the smallest place worth is , so multiplying never shrinks it below the face value.
In the units column, face value and place value are always equal.
True. The units place is worth , so place value face value face value. This is the only column where they match.
Two different digits can have the same place value in the same number.
False for the ordinary meaning of "place value of a digit," because each digit sits in its own column with its own power of ten — so their place values differ unless the digits are equal and the columns are equal, which cannot both happen for two different positions.
Moving a digit one column to the left multiplies its contribution by 10.
True. Each step left raises the power of ten by one (), so the digit's contribution to the number is multiplied by .
A larger digit always contributes more to the number than a smaller digit.
False. A tiny in the lakhs column () beats a big in the tens column (); the column decides the contribution, not the digit's size.
The number has a place value.
True — its place value is (face value times any power of ten is ). But its placeholder job of holding a column open is not a "value" at all; keep those two ideas separate.
Writing a number in expanded form and adding it back always returns the original number.
True. Expanded form is literally the sum of every column's contribution, so re-adding the pieces must rebuild the whole.
Spot the error
"In , the place value of is ."
Error: that is the face value. The sits in the tens column, so its place value is .
" in the Indian system is written ."
Error: that is the international 3-3-3 grouping. Indian grouping is 3 then 2-2, giving .
"Removing the from doesn't change anything because adds nothing."
Error: the adds to the sum, but it holds the hundreds column open. Delete it and collapses to — every digit slides right into the wrong column.
"The place value of in is , which equals six lakh, and that is also its face value in lakhs."
The place value () is correct, but calling it a "face value in lakhs" is the error — face value is the bare digit , with no column attached ever.
"To read I group it as from the left."
Error: Indian grouping starts from the right, giving — "three crore, seventy thousand, five." Grouping from the left scrambles the columns.
"Since lakh and million , a lakh and a million are the same kind of unit and interchangeable."
Error: they are different sizes ( million lakh). The Indian and international systems name the same underlying number differently — but a lakh is not a million.
"The digit in the ten-thousands place of is , so ten-thousands don't exist in this number."
Error: the column absolutely exists — it is just holding a . The empty column is what keeps up in the lakhs place; erase the idea of the column and the number breaks.
Why questions
Why can just 10 symbols write every number ever?
Because position carries extra information: the same digit means a different amount in a different column, so we recycle – endlessly instead of inventing a new symbol per quantity.
Why is each left column worth exactly 10 times the one on its right, and not, say, 7 times?
Because we group by tens (ten fingers): a column resets after and carries one, so exactly of one column bundle into of the next. See Number Systems — base and digits for other bases.
Why does the first Indian comma come after 3 digits but the rest after 2?
The 3 marks off units–tens–hundreds up to thousands; then lakhs and crores each span 2 digits, so 2-2 chopping lets you read the ladder units → thousands → lakhs → crores instantly.
Why is called a placeholder rather than just "nothing"?
Because it does a positional job: it keeps later digits parked in their correct high-value columns. Without it, columns slide and the whole number changes meaning.
Why does the expanded form have to be true, not just a trick?
Because a number literally is the total of its column contributions — each digit counts how many of the worth you have, so summing them is the definition of the number. See Expanded form and standard form.
Why does comparing two numbers usually start from the leftmost digit?
The leftmost non-zero digit sits in the highest-worth column, so it dominates the total; a difference there outweighs everything to its right. See Comparing and Ordering Numbers.
Why is the crores place and not ?
Counting columns from units (): units, tens, hundreds, thousands, ten-thousands, lakhs (), ten-lakhs (), crores () — the crore is the 8th column, hence the exponent . See Powers of Ten and Exponents.
Edge cases
What is the place value of every digit in a string of all zeros, like ?
Each has place value , and the whole thing is just the number — the columns exist but hold nothing, so there is no "leading" information at all.
Does a leading zero, like , change the number?
No. A leading zero sits in a column but holds , contributing nothing and holding no lower column open, so . Leading zeros are the only zeros you can safely delete.
What happens to place values when the number has no more digits — is there a "largest place"?
There is no largest place; you can always add another column worth the last (ten-crores, arab, and beyond). The ladder is infinite because you can always carry one more step left.
For a single-digit number like , how do face value and place value compare?
They are equal: a lone sits in the units place (), so place value face value. Single digits are the trivial case where the two ideas coincide.
Is the same number written (Indian) and (international) two different numbers?
No — it is one identical quantity, one lakh, wearing two different comma-costumes. Only the grouping and naming differ, never the value. See International vs Indian Number System.
If you round to the nearest thousand, does the place value idea change?
The idea is untouched — rounding only replaces lower-column digits with zeros (placeholders); each remaining digit keeps its column and its place value. See Rounding and Estimation.
Connections
- 1.1.02 Place value system — units, tens, hundreds, thousands, lakhs, crores (Hinglish)
- Number Systems — base and digits
- Expanded form and standard form
- Comparing and Ordering Numbers
- International vs Indian Number System
- Rounding and Estimation
- Powers of Ten and Exponents