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A Story of Numbers - Some Early Number Systems

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Egyptian Number System was an additive system using symbols for powers of 1010. For example, a single stroke ∣| represented 11, a heel bone ∩\cap represented 1010, and a coil of rope represented 100100.

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The Babylonian System was a positional system based on the number 6060 (sexagesimal). It used two primary symbols: a wedge ▼\blacktriangledown for 11 and a chevron ⟨\langle for 1010.

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Roman Numerals use seven basic symbols: I=1,V=5,X=10,L=50,C=100,D=500,M=1000I=1, V=5, X=10, L=50, C=100, D=500, M=1000. They follow specific additive and subtractive rules.

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In Roman Numerals, a smaller symbol placed after a larger one is added (e.g., VI=5+1=6VI = 5 + 1 = 6), whereas a smaller symbol placed before a larger one is subtracted (e.g., IV=5−1=4IV = 5 - 1 = 4).

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The Hindu-Arabic System is the modern decimal system (base 1010) which uses the digits 0,1,2,3,4,5,6,7,8,90, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is a positional system where the value of a digit depends on its place.

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The concept of 'Zero' (00) as a placeholder and a number was a revolutionary contribution of the Hindu-Arabic system, allowing for efficient representation of large numbers.

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Place Value: The value of a digit based on its position in a number. For example, in 456456, the place value of 55 is 5×10=505 \times 10 = 50.

📐Formulae

Place Value=Face Value×10nPlace\ Value = Face\ Value \times 10^{n} (where nn is the position starting from 00 at the units place)

Value=dn×10n+dn−1×10n−1+⋯+d1×101+d0×100Value = d_n \times 10^n + d_{n-1} \times 10^{n-1} + \dots + d_1 \times 10^1 + d_0 \times 10^0

Total (Roman)=∑Addition Rules−∑Subtraction RulesTotal\ (Roman) = \sum Addition\ Rules - \sum Subtraction\ Rules

💡Examples

Problem 1:

Convert the Roman numeral XCIXXCIX into the Hindu-Arabic system.

Solution:

XCIX=(100−10)+(10−1)=90+9=99XCIX = (100 - 10) + (10 - 1) = 90 + 9 = 99

Explanation:

In XCXC, XX (1010) is before CC (100100), so we subtract: 100−10=90100 - 10 = 90. In IXIX, II (11) is before XX (1010), so we subtract: 10−1=910 - 1 = 9. Adding them gives 9999.

Problem 2:

Find the difference between the place value and face value of the digit 77 in the number 8745287452.

Solution:

7000−7=69937000 - 7 = 6993

Explanation:

The face value of 77 is 77. Since 77 is in the thousands place, its place value is 7×1000=70007 \times 1000 = 7000. The difference is 7000−77000 - 7.

Problem 3:

Perform the subtraction of 45624562 from 90009000 using the vertical method.

Solution:

9000−45624438\begin{array}{r} 9000 \\ - 4562 \\ \hline 4438 \end{array}

Explanation:

We subtract each column starting from the units place, borrowing from the next higher place value where necessary.

Problem 4:

Write the Hindu-Arabic number 444444 in Roman Numerals.

Solution:

444=400+40+4=(500−100)+(50−10)+(5−1)=CD+XL+IV=CDXLIV444 = 400 + 40 + 4 = (500 - 100) + (50 - 10) + (5 - 1) = CD + XL + IV = CDXLIV

Explanation:

We break the number into hundreds, tens, and units. 400400 is written as CDCD, 4040 as XLXL, and 44 as IVIV.