1. Determine which numerals in the Hindu – Arabic numeration system are topologically equivalent.
2. Determine which letters in the English alphabet are topologically equivalent.
1. In the Hindu-Arabic numeration system, numerals can be categorized based on their topological equivalence, which means they have similar shapes. Here are the numerals that are topologically equivalent:
- 0 and 8: These numerals are topologically equivalent because they both form a closed loop.
- 6 and 9: These numerals are topologically equivalent because they both have a loop, but the orientation is different.
2. In the English alphabet, certain letters have similar shapes and can be considered topologically equivalent. Here are the letters that are topologically equivalent:
- O and Q: These letters are topologically equivalent because they both have a closed shape with a tail.
- B and D: These letters are topologically equivalent because they both have a circular shape with a straight line.
- M and W: These letters are topologically equivalent because they both consist of two upside-down Vs connected at the base.
It's important to note that topological equivalence is based on the visual similarity of the shapes, not the sound or usage of the letters or numerals.
To determine which numerals in the Hindu-Arabic numeration system are topologically equivalent, we need to look at the shapes of the numerals.
Here are the topologically equivalent groups of numerals in the Hindu-Arabic system:
1. Group 1: {0, 6, 8, 9} - These numerals have a closed loop. They can be rotated and reflected to form each other. For example, if you rotate a 6, it becomes a 9, and if you reflect an 8 vertically, it becomes a 9.
2. Group 2: {1, 7} - These numerals have a straight vertical line. They cannot be transformed into other numerals by any rotation or reflection.
3. Group 3: {2, 3, 4, 5} - These numerals are uniquely shaped and cannot be transformed into each other by rotation or reflection.
Now let's determine which letters in the English alphabet are topologically equivalent:
1. Group 1: {A, H, I, M, O, T, U, V, W, X, Y} - These letters have straight lines or combinations of straight lines. In some cases, rotation or reflection may transform one letter into another. For example, if you rotate an H or reflect it vertically, it becomes an I.
2. Group 2: {C, E, L, S} - These letters have curved shapes or combinations of curves. They cannot be transformed into other letters by rotation or reflection.
3. Group 3: {B, D, F, G, J, K, N, P, Q, R, Z} - These letters are uniquely shaped and cannot be transformed into each other by rotation or reflection.
It's important to note that topological equivalence considers the overall shape and structure of the numeral or letter, ignoring any specific details or variations in the font or handwriting.
12357
4690
8
The "4" depends on the font.
The way kids usually draw it, 4 belongs in the top row, since it has no hole.
Now do the letters