NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 65)#
To add numerals in a system with a base, we put together the symbols of each kind, then regroup: in the Egyptian system 10 of a symbol make one of the next, and in our base-5 system 5 of a symbol make one of the next. This is the same as carrying in column addition.
Egyptian symbols: stroke = 1, heel bone (∩) = 10, coil = 100, lotus = 1000, finger = 10,000.
Base-5 symbols: = 1, = 5, = 25, = 125, = 625.
1. Add the following Egyptian numerals: #
(i) (9 lotuses, 6 coils, 8 strokes) and (5 coils, 7 strokes)
Putting the symbols together:
- lotuses: 9
- coils: $6 + 5 = 11$
- strokes: $8 + 7 = 15$
Now we regroup, starting from the strokes:
- 15 strokes = 10 strokes + 5 strokes, and 10 strokes make 1 heel bone. That leaves 5 strokes and 1 heel bone.
- 11 coils = 10 coils + 1 coil, and 10 coils make 1 lotus. That leaves 1 coil, and the lotuses become 10.
- 10 lotuses make 1 finger, leaving no lotuses.
Sum = 1 finger, 1 coil, 1 heel bone, 5 strokes
Check in Hindu numerals: $9608 + 507 = 10115$, and $10115 = 10000 + 100 + 10 + 5$.
(ii) (1 lotus, 8 heel bones) and (4 heel bones, 6 strokes)
Putting the symbols together:
- lotuses: 1
- heel bones: $8 + 4 = 12$
- strokes: 6
12 heel bones = 10 heel bones + 2 heel bones, and 10 heel bones make 1 coil.
Sum = 1 lotus, 1 coil, 2 heel bones, 6 strokes
Check in Hindu numerals: $1080 + 46 = 1126$.
2. Add the following numerals that are in the base-5 system that we created: + #
Solution
Putting the symbols together:
- : $1 + 3 = 4$
- : $2 + 1 = 3$
- : $1 + 2 = 3$
- : $2 + 2 = 4$
No symbol occurs 5 or more times, so no regrouping is needed.
Sum =
Check in Hindu numerals:
stands for $125 + 50 + 5 + 2 = 182$
stands for $375 + 25 + 10 + 2 = 412$
$182 + 412 = 594$, and $4 \times 125 + 3 \times 25 + 3 \times 5 + 4 = 500 + 75 + 15 + 4 = 594$. ✓