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Expanded values
We usually use decimals for numbers: This means we have 10 digits that can go in one column. It is called base10, one column can be filled with ten options: (0-9,) before a new digit is added 10-19, 20-29 etc
Hexadecimal are a base16 number system. One column can hold sixteen units, before a second column is added.(0-15)
Computers use a binary number system: base2, each column can have a 1 or 0 bit, anymore and another column is added. Eg 0 in decimal = 0 binary, 1 in decimal = 1 in binary, 2 in decimal = 10 in binary 3 in decimal = 11 in binary
When converting between different number systems expanding the values can help make sense of what the value is in each column. We translate expanded values in decimal usually, this is because we count in 10s most often and it is easier for us to understand.
111 in different number systems
| n2 n*n | n1 n | n0 1 | Expanded value | Decimal | ||
| 3rd column | 2nd column | 1st column | 100s | 10s | 1s | |
| 1 | 1 | 1 | Base1= 1 + 1 + 1 = | 0 | 0 | 3 |
| 1 | 1 | 1 | Base2= 4 + 2 + 1 = | 0 | 0 | 7 |
| 1 | 1 | 1 | Base10= 100 + 10 + 1 = | 1 | 1 | 1 |
| 1 | 1 | 1 | Base16= 256 + 16 + 1 = | 2 | 7 | 3 |
Unary - Base1
| 12 1*1 | 11 1 | 10 1 | Expanded value of UNARY | Decimal | |
| 1s | 1s | 1s | 10s | 1s | |
| 0 | 0 | 1 | 0 + 0 + 1 | 0 | 1 |
| 0 | 1 | 1 | 0 + 0 + 1 | 0 | 2 |
| 1 | 1 | 1 | 1 + 1 + 1 | 0 | 3 |
Decimal - Base10
| 102 (10*10) | 101 (10*1) | 100 (1) | Expanded value of Decimal | Written numerals |
| 100s | 10s | 1s | ||
| 0 | 0 | 1 | (100*0) + (10*0) + (1*1) | One |
| 0 | 1 | 1 | (100*0) + (10*1) + (1*1) | Eleven |
| 1 | 1 | 1 | (100*1) + (10*1) + (1*1) | One Hundred and Eleven |
| 0 | 0 | 0 | (100*0) + (10*0) + (1*0) | Zero |
| 0 | 0 | 5 | (100*0) + (10*0) + (1*5) | Five |
| 0 | 5 | 0 | (100*0) + (10*5) + (1*0) | Fifty |
| 0 | 9 | 4 | (100*0) + (10*9) + (1*4) | Ninety four |
| 1 | 2 | 3 | (100*1) + (10*2) + (1*3) | One Hundred and Twenty Three |
| 6 | 7 | 9 | (100*6) + (10*7) + (1*9) | Six Hundred and Seventy Nine |
Hexadecimal - Base16
| Hex | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | 10 | 11 | 12 | 13 |
| Dec | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 162 16*16 | 161 16 | 160 1 | Expanded value of Hexadecimal | Decimal | ||
| 256s | 16s | 1s | 100s | 10s | 1s | |
| 0 | 0 | 1 | (256*0) + (16*0) + (1*1) | 0 | 0 | 1 |
| 0 | 1 | 1 | (256*0) + (16*1) + (1*1) | 0 | 1 | 7 |
| 1 | 1 | 1 | (256*1) + (16*1) + (1*1) | 2 | 6 | 3 |
| 0 | 0 | 1 | (256*0) + (16*0) + (1*1) | 0 | 0 | 1 |
| 0 | 0 | 9 | (256*0) + (16*0) + (1*9) | 0 | 0 | 9 |
| 0 | 0 | A | (256*0) + (16*0) + (1*10) | 0 | 1 | 0 |
| 0 | 0 | F | (256*0) + (16*0) + (1*15) | 0 | 1 | 5 |
| 0 | 1 | 0 | (256*0) + (16*1) + (1*0) | 0 | 1 | 6 |
| 0 | 1 | 2 | (256*0) + (16*1) + (1*2) | 0 | 1 | 8 |
| 1 | 3 | 3 | (256*1) + (16*3) + (1*3) | 3 | 0 | 7 |
| 1 | 2 | 3 | (256*1) + (16*2) + (1*3) | 2 | 9 | 1 |
| 3 | 3 | 3 | (256*3) + (16*3) + (1*3) =
| 8 | 9 | 1 |
Binary - Base2
| 23 2*2*2 | 22 2*2 | 21 2 | 20 1 | Expanded value of Binary | Decimal | |
| 8s | 4s | 2s | 1s | 10s | 1s | |
| 0 | 0 | 0 | 0 | (8*0) + (4*0) + (2*0) + (1*0) | 0 | 0 |
| 0 | 0 | 0 | 1 | (8*0) + (4*0) + (2*0) + (1*1) | 0 | 1 |
| 0 | 0 | 1 | 0 | (8*0) + (4*0) + (2*1) + (1*0) | 0 | 2 |
| 0 | 0 | 1 | 1 | (8*0) + (4*0) + (2*1) + (1*1) | 0 | 3 |
| 0 | 1 | 0 | 0 | (8*0) + (4*1) + (2*0) + (1*0) | 0 | 4 |
| 0 | 1 | 0 | 1 | (8*0) + (4*1) + (2*0) + (1*1) | 0 | 5 |
| 0 | 1 | 1 | 0 | (8*0) + (4*1) + (2*1) + (1*0) | 0 | 6 |
| 0 | 1 | 1 | 1 | (8*0) + (4*1) + (2*1) + (1*1) | 0 | 7 |
| 1 | 0 | 0 | 0 | (8*1) + (4*0) + (2*0) + (1*0) | 0 | 8 |
| 1 | 1 | 1 | 1 | (8*1) + (4*1) + (2*1) + (1*1) | 1 | 5 |