Number Base worksheet

1. In binary: 11001100110

1. Multiply each digit by its place value, a power of 23, counting powers from 0 at the right

   $3\times {23}^2+2\times {23}^1+5\times {23}^0$$=\mathrm{1,587}+46+5$

2. Add the place values to get the decimal value

   $\mathrm{1,587}+46+5$$=\mathrm{1,638}$

3. Divide by 2: quotient 819, remainder 0

   $\mathrm{1,638}\text{÷}2\text{=}819\text{remainder}0$

4. Divide by 2: quotient 409, remainder 1

   $819\text{÷}2\text{=}409\text{remainder}1$

5. Divide by 2: quotient 204, remainder 1

   $409\text{÷}2\text{=}204\text{remainder}1$

6. Divide by 2: quotient 102, remainder 0

   $204\text{÷}2\text{=}102\text{remainder}0$

7. Keep dividing the quotient by 2 until it reaches 0 (11 divisions in all; the last two are shown)

8. Divide by 2: quotient 1, remainder 1

   $3\text{÷}2\text{=}1\text{remainder}1$

9. Divide by 2: quotient 0, remainder 1

   $1\text{÷}2\text{=}0\text{remainder}1$

10. Read the remainders from the last one to the first to get the digits

    $\text{11001100110}$

11. Write the pattern in binary, padded with leading zeros to 16 bits

    $\text{0000 0110 0110 0110}$

2. In base 15: 6AC

1. Multiply each digit by its place value, a power of 27, counting powers from 0 at the right

   $2\times {27}^2+2\times {27}^1$$=\mathrm{1,458}+54$

2. Add the place values to get the decimal value

   $\mathrm{1,458}+54$$=\mathrm{1,512}$

3. Divide by 15: quotient 100, remainder 12 (digit C)

   $\mathrm{1,512}\text{÷}15\text{=}100\text{remainder}12$

4. Divide by 15: quotient 6, remainder 10 (digit A)

   $100\text{÷}15\text{=}6\text{remainder}10$

5. Divide by 15: quotient 0, remainder 6

   $6\text{÷}15\text{=}0\text{remainder}6$

6. Read the remainders from the last one to the first to get the digits

   $\text{6AC}$

7. Write the pattern in binary, padded with leading zeros to 64 bits

   $\text{0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 1110 1000}$

3. In base 4: 1332

1. Multiply each digit by its place value, a power of 7, counting powers from 0 at the right

   $2\times 7^2+4\times 7^1$$=98+28$

2. Add the place values to get the decimal value

   $98+28$$=126$

3. Divide by 4: quotient 31, remainder 2

   $126\text{÷}4\text{=}31\text{remainder}2$

4. Divide by 4: quotient 7, remainder 3

   $31\text{÷}4\text{=}7\text{remainder}3$

5. Divide by 4: quotient 1, remainder 3

   $7\text{÷}4\text{=}1\text{remainder}3$

6. Divide by 4: quotient 0, remainder 1

   $1\text{÷}4\text{=}0\text{remainder}1$

7. Read the remainders from the last one to the first to get the digits

   $\text{1332}$

8. Write the pattern in binary, padded with leading zeros to 16 bits

   $\text{0000 0000 0111 1110}$

4. In base 21: 32A

1. Multiply each digit by its place value, a power of 25, counting powers from 0 at the right

   $2\times {25}^2+5\times {25}^1$$=\mathrm{1,250}+125$

2. Add the place values to get the decimal value

   $\mathrm{1,250}+125$$=\mathrm{1,375}$

3. Divide by 21: quotient 65, remainder 10 (digit A)

   $\mathrm{1,375}\text{÷}21\text{=}65\text{remainder}10$

4. Divide by 21: quotient 3, remainder 2

   $65\text{÷}21\text{=}3\text{remainder}2$

5. Divide by 21: quotient 0, remainder 3

   $3\text{÷}21\text{=}0\text{remainder}3$

6. Read the remainders from the last one to the first to get the digits

   $\text{32A}$

7. Write the pattern in binary, padded with leading zeros to 32 bits

   $\text{0000 0000 0000 0000 0000 0101 0101 1111}$

5. In base 7: 10362

1. Multiply each digit by its place value, a power of 36, counting powers from 0 at the right

   $2\times {36}^2$$=\mathrm{2,592}$

2. Add the place values to get the decimal value

   $\mathrm{2,592}$$=\mathrm{2,592}$

3. Divide by 7: quotient 370, remainder 2

   $\mathrm{2,592}\text{÷}7\text{=}370\text{remainder}2$

4. Divide by 7: quotient 52, remainder 6

   $370\text{÷}7\text{=}52\text{remainder}6$

5. Divide by 7: quotient 7, remainder 3

   $52\text{÷}7\text{=}7\text{remainder}3$

6. Divide by 7: quotient 1, remainder 0

   $7\text{÷}7\text{=}1\text{remainder}0$

7. Divide by 7: quotient 0, remainder 1

   $1\text{÷}7\text{=}0\text{remainder}1$

8. Read the remainders from the last one to the first to get the digits

   $\text{10362}$

9. Write the pattern in binary, padded with leading zeros to 32 bits

   $\text{0000 0000 0000 0000 0000 1010 0010 0000}$

6. In base 7: 6303

1. Multiply each digit by its place value, a power of 32, counting powers from 0 at the right

   $2\times {32}^2+5\times {32}^1$$=\mathrm{2,048}+160$

2. Add the place values to get the decimal value

   $\mathrm{2,048}+160$$=\mathrm{2,208}$

3. Divide by 7: quotient 315, remainder 3

   $\mathrm{2,208}\text{÷}7\text{=}315\text{remainder}3$

4. Divide by 7: quotient 45, remainder 0

   $315\text{÷}7\text{=}45\text{remainder}0$

5. Divide by 7: quotient 6, remainder 3

   $45\text{÷}7\text{=}6\text{remainder}3$

6. Divide by 7: quotient 0, remainder 6

   $6\text{÷}7\text{=}0\text{remainder}6$

7. Read the remainders from the last one to the first to get the digits

   $\text{6303}$

8. Write the pattern in binary, padded with leading zeros to 64 bits

   $\text{0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 1010 0000}$

7. In base 34: 3EB

1. Multiply each digit by its place value, a power of 35, counting powers from 0 at the right

   $3\times {35}^2+8\times {35}^1$$=\mathrm{3,675}+280$

2. Add the place values to get the decimal value

   $\mathrm{3,675}+280$$=\mathrm{3,955}$

3. Divide by 34: quotient 116, remainder 11 (digit B)

   $\mathrm{3,955}\text{÷}34\text{=}116\text{remainder}11$

4. Divide by 34: quotient 3, remainder 14 (digit E)

   $116\text{÷}34\text{=}3\text{remainder}14$

5. Divide by 34: quotient 0, remainder 3

   $3\text{÷}34\text{=}0\text{remainder}3$

6. Read the remainders from the last one to the first to get the digits

   $\text{3EB}$

7. Write the pattern in binary, padded with leading zeros to 32 bits

   $\text{0000 0000 0000 0000 0000 1111 0111 0011}$

8. In base 32: 78

1. Multiply each digit by its place value, a power of 8, counting powers from 0 at the right

   $3\times 8^2+5\times 8^1$$=192+40$

2. Add the place values to get the decimal value

   $192+40$$=232$

3. Divide by 32: quotient 7, remainder 8

   $232\text{÷}32\text{=}7\text{remainder}8$

4. Divide by 32: quotient 0, remainder 7

   $7\text{÷}32\text{=}0\text{remainder}7$

5. Read the remainders from the last one to the first to get the digits

   $\text{78}$

6. Write the pattern in binary, padded with leading zeros to 8 bits

   $\text{1110 1000}$

9. In base 31: 1UT

1. Multiply each digit by its place value, a power of 30, counting powers from 0 at the right

   $2\times {30}^2+4\times {30}^1$$=\mathrm{1,800}+120$

2. Add the place values to get the decimal value

   $\mathrm{1,800}+120$$=\mathrm{1,920}$

3. Divide by 31: quotient 61, remainder 29 (digit T)

   $\mathrm{1,920}\text{÷}31\text{=}61\text{remainder}29$

4. Divide by 31: quotient 1, remainder 30 (digit U)

   $61\text{÷}31\text{=}1\text{remainder}30$

5. Divide by 31: quotient 0, remainder 1

   $1\text{÷}31\text{=}0\text{remainder}1$

6. Read the remainders from the last one to the first to get the digits

   $\text{1UT}$

10. In base 25: B0

1. Multiply each digit by its place value, a power of 11, counting powers from 0 at the right

   $2\times {11}^2+3\times {11}^1$$=242+33$

2. Add the place values to get the decimal value

   $242+33$$=275$

3. Divide by 25: quotient 11, remainder 0

   $275\text{÷}25\text{=}11\text{remainder}0$

4. Divide by 25: quotient 0, remainder 11 (digit B)

   $11\text{÷}25\text{=}0\text{remainder}11$

5. Read the remainders from the last one to the first to get the digits

   $\text{B0}$

6. Write the pattern in binary, padded with leading zeros to 32 bits

   $\text{0000 0000 0000 0000 0000 0001 0001 0011}$