स्थानीयमानPlace Value
Study notes and preparation material for Place Value under Mathematics section.
स्थानीयमान एवं जातीयमान का परिचयIntroduction to Place Value & Face Value
किसी संख्या में प्रत्येक अंक का मान उसके स्थान (स्थिति) पर निर्भर करता है। दाईं ओर से बाईं ओर बढ़ते हुए प्रत्येक स्थान का मान पिछले स्थान से 10 गुना होता है: इकाई (1), दहाई (10), सैकड़ा (100), हज़ार (1000), दस हज़ार (10000), आदि। In a decimal base-10 positional numeral system, each digit's value is determined by its position. Moving from right to left, each place is 10 times larger than the previous: ones (1), tens (10), hundreds (100), thousands (1000), and ten-thousands (10,000).
| तत्व (Aspect) | जातीयमान / अंकित मान (Face Value) | स्थानीयमान (Place Value) |
|---|---|---|
| परिभाषा (Definition) | अंक का वास्तविक मान, जो कभी नहीं बदलता। | अंक का स्थितिगत मान = अंक × स्थान मान। |
| संख्या 7,532 में अंक 5 | 5 | 5 × 100 = 500 (सैकड़ा) |
| शून्य (0) का मान | 0 | सदैव 0 (किसी भी स्थान पर 0 × स्थान = 0) |
Formula and skill map
Base-ten place values
From right to left: ones, tens, hundreds, thousands, ten-thousands, and so on. Each place is ten times the place to its right. A digit’s value is digit × place value.
Reading, rounding and notation
Face value is the digit itself; place value depends on position. Indian grouping uses lakh/crore; international grouping uses million/billion. Round by checking the digit immediately to the right of the target place.
Exam checks
Read the whole number aloud to check its grouping.
Use expanded form when a digit’s value is unclear.
Use expanded form when a digit’s value is unclear.
Core concepts and worked examples
Face value and place value
Worked example: in 5,47,326, the face value of 7 is 7; its place value is 7,000 because it is in the thousands place.
Expanded form
Worked example: 8,03,052 = \(8\times100000+3\times1000+5\times10+2\). Zero placeholders mark empty places.
Compare and order numbers
Compare the number of digits first, then the leftmost unequal digit. Worked example: 5,08,900 > 4,99,999 because the lakh digit is larger.
Regrouping and carrying
Ten units of a place make one unit of the next. Worked example: 2,758 + 1,486 = 4,244 after regrouping ones, tens and hundreds.
Rounding numbers
Worked example: round 7,46,582 to nearest thousand. Hundreds digit is 5, so round up to 7,47,000.
Indian and international grouping
The same number may be written 12,34,567 (Indian) or 1,234,567 (international); value does not change. Read the named grouping carefully. Worked example: 12,34,567 and 1,234,567 denote the same value: one million two hundred thirty-four thousand five hundred sixty-seven.
Question types and solving approaches
Find the value of a digit
Worked example: in 63,418, the 3 has value 3,000 and face value 3.
Write expanded or standard form
Worked example: 4,00,000 + 20,000 + 600 + 9 = 4,20,609.
Compare or arrange numbers
Compare from the highest place; zeros are meaningful placeholders. Worked example: 6,020 > 5,999 because 6 thousands exceed 5 thousands.
Round to a stated place
Mark the target digit, inspect the next digit, then replace all lower places by zeros. Worked example: 4,786 rounded to the nearest hundred is 4,800.
Use place value in calculations
Check each regrouped unit remains equivalent: 10 tens become 1 hundred. Worked example: 368 + 257 = 625; regroup 15 ones as 1 ten and 5 ones.
Proofs and derivations
Why each place is ten times the next
In base ten, a digit’s value is multiplied by \(10^k\) at position \(k\). Moving one place left multiplies its value by 10.
Why a zero placeholder matters
Zero contributes no quantity in its own place but keeps digits in their correct positions; removing it may change the number by a factor of ten or more.
Why rounding uses the next digit
The next digit determines whether the value is closer to the target multiple or the next one; digits farther right do not change that nearest-place decision.
Common errors and how to avoid them
Confusing face and place value
Face value is the digit; place value includes its position.
Dropping placeholder zeros
Keep zeros needed to preserve positions in standard form.
Rounding the wrong digit
Look one place to the right of the rounding place.
Mixing number grouping systems
Use separators and names consistently; commas do not change numerical value.