3.5 Going Deeper: Enrichment & Exam Preparation

Where Number Systems Show Up in Real Life

Math in the Real World

  • Computers think in binary (base 2). Every photo, song and message inside a phone is ultimately a string of 0s and 1s, because an electronic switch has just two states, off (0) and on (1). The number 45 travels through a circuit as 101101_{(2)}; the colour “pure red” on a screen is stored as 255 = 11111111_{(2)}.
  • Hexadecimal (base 16) for colours and memory. Web designers write colours as codes like #FF0000. Here FF is hexadecimal for 255, the largest one-byte value. Base 16 is popular because four binary digits fold neatly into one hex digit.
  • Clocks and angles still run on base 60. The Mesopotamian sexagesimal system never died: 1 hour = 60 minutes, 1 minute = 60 seconds, and a full turn is 360 = 6\times 60 degrees. When you say “quarter past three” you are doing base-60 arithmetic.
  • Place value is why the column method works. Adding 476 + 358 by carrying is nothing more than bundling ten ones into a ten, ten tens into a hundred, the Hindu place-value idea in action every single day.
  • Roman numerals survive as labels. Clock faces, book chapters, Olympic Games (Paris 2024 = the Games of the XXXIII Olympiad), and movie sequels still use I, V, X, L, C, D, M, a 2000-year-old landmark-number scheme.

Quick Reference: Numerals, Bases & Place Value

Keeping these tables in your head turns slow conversions into instant recall under exam pressure.

Symbol I V X L C D M
Value 1 5 10 50 100 500 1000
Subtractive form IV = 4 IX = 9 XL = 40 XC = 90 CD = 400 CM = 900 ,

Roman rules in one line: a smaller symbol after a larger one adds (VI = 6); a smaller symbol before a larger one subtracts (IV = 4). Never repeat a symbol more than three times in a row, and only I, X, C are used to subtract.

Place Indian name International name Power of 10
1 ones ones 10^0
10 tens tens 10^1
100 hundreds hundreds 10^2
1,000 thousands thousands 10^3
10,000 ten thousands ten thousands 10^4
1,00,000 lakh hundred thousands 10^5
10,00,000 ten lakh million 10^6
1,00,00,000 crore ten million 10^7

The same number 7654321 is grouped 76,\!54,\!321 in the Indian system (76 lakh 54 thousand 321) but 7,\!654,\!321 in the International system (7 million 654 thousand 321), the commas fall in different places.

To convert Method Example
base 10 \to base n Divide repeatedly by n; read remainders bottom-up 29 \to 35_{(6)}
base n \to base 10 Multiply each digit by its power of n and add 1342_{(6)} = 350
base 10 \to base 2 Repeatedly halve, record remainders 45 \to 101101_{(2)}
Roman \to number Add values; subtract when a smaller precedes a larger XLII = 42

Use the converter below to turn any whole number into a Roman numeral and back again, and to check the division-remainder method for yourself.

Roman numeral ↔ number converter

Memory Tricks & One-Page Revision

Quick Revision Card

  • Four great ideas, in order: grouping \to landmark numbers \to a base (powers of n) \to place value (with 0).
  • Landmark numbers of base n are 1, n, n^2, n^3, \dots (base 10 \Rightarrow 1,10,100,1000).
  • base 10 \to base n: divide by n again and again; the remainders read bottom-to-top give the digits.
  • base n \to base 10: multiply each digit by its place value (a power of n) and add.
  • Roman shortcut: “smaller before larger means subtract.” IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900.
  • Indian grouping: ones, tens, hundreds, then commas every two digits, thousand, lakh, crore. International: commas every three, thousand, million, billion.
  • Zero does two jobs: a placeholder (holds an empty position) and a number (obeys 0+a=a, 0\times a=0). The Indian system was first to treat 0 as both.

Spot the Mistake

Common Exam Mistakes

  • Reading remainders top-to-bottom when converting base 10 to another base, they must be read bottom-to-top. (29 \div 6: remainders 5 then 4, so the answer is 45_{(6)}, not 54_{(6)}.)
  • Forgetting the subtractive rule and writing 4 as IIII or 9 as VIIII instead of IV and IX.
  • Treating a digit’s value as fixed. In 202 the two 2s are not equal, the left one is worth 200, the right one worth 2. That is the whole point of place value.
  • Dropping a needed 0 in the middle of a number: writing 407 as 47. The 0 is a real digit holding the tens place.
  • Putting the commas in the wrong grouping, using International grouping while reading an Indian-named number (or vice versa) gives the wrong name (lakh vs hundred-thousand).

Test Your Speed

A quick drill: a base-conversion or Roman-numeral question appears, you type the answer and build a streak.

Numeral sprint
Convert and press Check.
…
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. The Roman numeral for 42 is ______.
  2. The binary numeral 1011_{(2)} stands for the base-10 number ______.
  3. In the number 4709, the digit 7 has place value ______.

Short answer (2 marks each)

  1. Convert 73 (base 10) into base 2.
  2. Write 156 (base 10) in base 8, showing the division-remainder steps.

Long answer (3 marks each)

  1. Convert the base-6 numeral 1342_{(6)} into base 10, and then rewrite that value in base 8.
  2. Write 50709 in expanded form using powers of 10, and name it in both the Indian and the International systems.

HOTS (Higher Order Thinking)

  1. What is the largest three-digit number in base 6, and what is its value in base 10?
  2. Explain why the Egyptian system needed endless new symbols for large numbers, while the Hindu place-value system needs only ten. Which single idea makes the difference?

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): In a base-n system, no digit can be n or larger.   Reason (R): Once you collect n copies of a landmark number, they bundle into one copy of the next landmark.
  1. 42 = 40 + 2 = \text{XL} + \text{II} = \mathbf{XLII}.
  2. 1011_{(2)} = 1\times 8 + 0\times 4 + 1\times 2 + 1 = \mathbf{11}.
  3. The 7 sits in the hundreds place, so its place value is \mathbf{700}.
  4. Halve repeatedly: 73\to 36\,r1,\;18\,r0,\;9\,r0,\;4\,r1,\;2\,r0,\;1\,r0,\;0\,r1. Reading the remainders bottom-up gives \mathbf{1001001}_{(2)}. (Check: 64+8+1=73.)
  5. 156 \div 8 = 19\,r4, then 19 \div 8 = 2\,r3, then 2 \div 8 = 0\,r2. Bottom-up: \mathbf{234}_{(8)}. (Check: 2\times 64 + 3\times 8 + 4 = 128+24+4 = 156.)
  6. 1342_{(6)} = 1\times 216 + 3\times 36 + 4\times 6 + 2 = 216 + 108 + 24 + 2 = \mathbf{350}. Then 350 in base 8: 350\div 8 = 43\,r6,\;43\div 8 = 5\,r3,\;5\div 8 = 0\,r5, so 350 = \mathbf{536}_{(8)}. (Check: 5\times 64 + 3\times 8 + 6 = 320+24+6 = 350.)
  7. 50709 = 5\times 10^4 + 0\times 10^3 + 7\times 10^2 + 0\times 10 + 9\times 1. Indian: fifty thousand seven hundred nine (50,\!709). International: fifty thousand seven hundred nine (50,\!709), for this number both systems read alike, but the comma grouping is identical only up to the thousand; the difference shows from a lakh upward.
  8. The largest three-digit base-6 number is 555_{(6)} (the biggest digit is 5). Its value is 5\times 36 + 5\times 6 + 5 = 180 + 30 + 5 = \mathbf{215}.
  9. The Egyptian scheme drew a new picture-symbol for every power of 10 (1, 10, 100, \dots), so bigger numbers forever demand fresh symbols. The Hindu system instead uses place value: the position of a digit tells you which power of 10 it counts, so the same ten symbols 0–9 describe every number. Place value (with 0) is the deciding idea.
  10. (a), Both statements are true and R explains A: a digit reaching n would mean n copies of a landmark, which always regroup into one of the next landmark, so digits only run 0 to n-1.

Connections to Other Chapters

How This Chapter Links Forward

  • Chapter 1 (A Square and A Cube): the powers n^2, n^3 that name squares and cubes are exactly the landmark numbers of a base-n system.
  • Chapter 2 (Power Play): place value is an exponent idea, each digit is multiplied by a power of 10, leading straight into exponents and scientific notation.
  • Computer science & later study: base 2 (binary) and base 16 (hexadecimal) are the everyday languages of computing, built on the very same grouping-and-place-value ideas met here.

Glossary

Key Terms

  • Numeral: a written symbol (or string of symbols) standing for a number, e.g. 7, 36, IX.
  • One-to-one mapping: pairing each object with exactly one item of a standard sequence, the engine of counting.
  • Landmark number: a reference value that earns its own symbol or place (Roman I, V, X; or the powers of a base).
  • Base (n): the fixed multiplier whose powers 1, n, n^2, \dots are the landmark numbers of the system.
  • Place value (positional system): a scheme in which a digit’s worth depends on its position, e.g. the two 2s in 202.
  • Placeholder: a symbol (such as 0) marking an empty position so that positions are not misread.
  • Zero as a number: treating 0 as a full number obeying 0+a=a and 0\times a=0, the decisive Indian contribution.