Page 28 - Start Up Mathematics_6

Expanding brackets
We will now see how the use of brackets allows us to make calculations simple and systematic.
(i) 13 × 1,013 = 13 × (1,000 + 13)
= 13 × 1,000 + 13 × 13
= 13,000 + 169 = 13,169
(ii) 102 × 105 = 102 × (100 + 5)
= 102 × 100 + 102 × 5
= (100 + 2) × 100 + (100 + 2) × 5
= 100 × 100 + 2 × 100 + 100 × 5 + 2 × 5
= 10,000 + 200 + 500 + 10 = 10,710

Roman Numerals
We are well aware of the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. These numerals are used in Hindu-
Arabic numeral system. However, one of the old systems of numeration developed by Romans is
also in common use. This system is called Roman system of numeration. There are seven distinct
numeral symbols in this system. These symbols and their corresponding Hindu-Arabic numerals
are given below:

Roman Numerals I V X L C D M
Hindu-Arabic Numerals 1 5 10 50 100 500 1,000

Rules followed in the Roman System of Numeration
Rule 1: If a symbol is repeated, its value is added as many times as it occurs.
Examples:
 III = 1 + 1 + 1 = 3  XX = 10 + 10 = 20

 CC = 100 + 100 = 200  MMM = 1,000 + 1,000 + 1,000 = 3,000
Remarks:
(i) A symbol can be repeated at the most three times. Do you know?
(ii) Symbols V, L and D are never repeated. In Roman system of numeration,
there is no symbol for zero.
(iii) The only symbols which can repeat are I, X, C and M.

Rule 2: A symbol of smaller value written to the right of a symbol of greater value always gets
added to the symbol of greater value.
Examples:
 VI = 5 + 1 = 6  VII = 5 + 1 + 1 = 7
 XI = 10 + 1 = 11  XIII = 10 + 1 + 1 + 1 = 13
 LX = 50 + 10 = 60  CXI = 100 + 10 + 1 = 111
Rule 3: A symbol of smaller value written to the left of a symbol of greater value always gets
subtracted from the symbol of greater value.

Examples:
 IV = 5 – 1 = 4  IX = 10 – 1 = 9
 XL = 50 – 10 = 40  XC = 100 – 10 = 90
 CD = 500 – 100 = 400  CM = 1,000 – 100 = 900

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