Number Bases: Yuck, You're Counting Wrong!

Alright, listen up, you lot! Number bases. It's how many symbols you're stuck with. Base 10? Ten symbols 0 to 9. They clump together, like, 12, 99, or that revolting 386.

Binary? Base 2. Two symbols. 0 and 1. Like a pair of dirty socks, that's all you get.

Hexadecimal, base 16, adds letters! A to F. Because why not? More junk to remember!

A = 10

B = 11

C = 12

D = 13

E = 14

F = 15

That tiny number at the bottom? That's your base. 9₁₀, base 10. 1011₂, base 2. 28₁₆, you got it. It's like a secret code, but not a very good one.

Base Conversion: Ugh, Do We Have To?

Want to change a binary or hexadecimal number? Fine. Each digit gets multiplied by the base, raised to a power. Starting from zero, naturally.

The Table of Mathematical Misery
Digit 4	Digit 3	Digit 2	Digit 1	Digit 0
× base⁴	× base³	× base²	× base¹	× base⁰

5²= 5 × 5. Duh. 5³ = 5 × 5 × 5. It's like chewing your own socks, pointless.

Worked examples

a) Convert the binary number 1010 to decimal.

Step 1: put the binary number into the table format as shown in Table above.

1	0	1	0
× base³	× base²	× base¹	× base⁰

Step 2: binary is base 2, so replace the word 'base' with 2.

1	0	1	0
× 2³	× 2²	× 2¹	× 2⁰

Step 3: calculate each column.

1 × 2³ = 8

0 × 2² = 0

1 × 2¹ = 2

0 × 2⁰ = 0

Step 4: add the columns together.

8 + 0 + 2 + 0 = 10

b) Convert the hexadecimal number 2C5 to decimal.

Step 1: put the hexadecimal number into the table format as shown above.

2	C	5
× base²	× base¹	× base⁰

Step 2: hexadecimal is base 16, so replace the word 'base' with 16.

2	C	5
× 16²	× 16¹	× 16⁰

Step 3: calculate each column, replacing any letters with their numerical equivalent. (C = 12; see box on previous page.)

2 × 16² = 512

12 × 16¹ = 192

5 × 16⁰ = 5

Step 4: add the columns together.

512 + 192 + 5 = 709

Guided Practice

Copy out the workings and complete the answers on a separate piece of paper.

1 Convert the binary number 11011 to denary. Show your working.

Steps 1 and 2: put the number into table format and fill in the number base for binary in each column. One has been done for you.

1	1	0	1	1
× 2⁴	× __³	× __²	× __¹	× __⁰

Steps 3 and 4: complete the calculation for each column and add the columns. Again, one has been done for you.

16 + __ + __ + __ + __

The answer is 27

2 Convert the hexadecimal number 2A to denary. Show your working.

Steps 1 and 2: put the number in table format and fill in the second hexadecimal number on the top row and the bases on the bottom row.

2	__
× __¹	× __⁰

Steps 3 and 4: complete the calculation for each column and add the columns.

__ + __

The answer is 42

Independent Practice

Convert the binary number 1101 to decimal.
Convert the hexadecimal number 3A to decimal.
Convert the binary number 10011 to decimal.
Convert the hexadecimal number C5 to decimal.
Convert the binary number 11010101 to decimal.
Convert the hexadecimal number 1E2F to decimal.

Answers

13
58
19
197
213
7727

Base 10 to Binary or Hexadecimal: Division? Seriously?

So you want to turn a regular number (base 10, the boring one) into binary or hexadecimal. Fine. Division. It's like chopping up your dinner, but with numbers.

Want base 2? Divide by 2. Keep the remainders. Want base 16? Divide by 16. Same deal with the remainders. Stack 'em up, read 'em backwards.

Remember, those remainders? If you're doing hexadecimal, and you get 10, 11, 12, 13, 14, or 15, you're not writing those down. No, no, no. You're using letters. A, B, C, D, E, and F.

A = 10

B = 11

C = 12

D = 13

E = 14

F = 15

It's all about those remainders. And don't forget, write them backwards. You're welcome. Or not.

Worked examples

Example 1: Convert 45₁₀ to base 2 (binary).

Divide the number by the base you want it in until you get 0. You want base 2 so you divide by 2.

Step 1: divide the number by 2.

45 ÷ 2 = 22 r 1

Step 2: take the quotient (the result) and divide it by 2.

22 ÷ 2 = 11 r 0

Step 3: repeat this process until you have 0—quotient divided by 2.

11 ÷ 2 = 5 r 1

5 ÷ 2 = 2 r 1

2 ÷ 2 = 1 r 0

1 ÷ 2 = 0 r 1

Make sure you write your remainder, as this will be used to make the answer.

Step 4: now you have 0, write all the remainders from the last to the first; 101101.

Example 2: Convert 75₁₀ to binary (base 2).

Step 1: divide 75 by 2.

75 ÷ 2 = 37 r 1

Step 2: divide the quotient by 2.

37 ÷ 2 = 18 r 1

Step 3: repeat until you have 0.

18 ÷ 2 = 9 r 0

9 ÷ 2 = 4 r 1

4 ÷ 2 = 2 r 0

2 ÷ 2 = 1 r 0

1 ÷ 2 = 0 r 1

Continue dividing by 2 until you have 0.

Step 4: write all the remainders, from the last to the first, 1001011.

Example 3: Convert 38₁₀ to base 16 (hexadecimal).

Step 1: divide the number by 16.

38 ÷ 16 = 2 r 6

Step 2: divide the quotient by 16.

2 ÷ 16 = 0 r 2

Step 3: repeat until you have 0.

Step 4: write the remainders, from the last to first; 2 6.

Step 5: check for numbers over 9. There are none, so 26 is the final answer.

Example 4: Convert 120₁₀ to hexadecimal (base 16).

Step 1: divide the number by 16.

120 ÷ 16 = 7 r 8

Step 2: divide the quotient by 16.

7 ÷ 16 = 0 r 7

Step 3: repeat until you have 0.

Step 4: write the remainders, from the last to first, 78.

Step 5: check for numbers over 9. There are none, so 78 is the final answer.

Guided Practice

Copy out the workings and complete the answers on a separate piece of paper.

1 Convert 73₁₀ to binary (base 2).

Step 1: divide the number by 2. 73 ÷ 2 = 36 r 1

Step 2: divide the quotient by 2. 36 ÷ 2 = 18 r __

Step 3: repeat until you have 0.

18 ÷ 2 = __ r __

__ ÷ 2 = __ r __

Step 4: write the remainders from last to first.

Answer: 1001001

2 Convert 198₁₀ to binary.

Step 1: divide the number by 2. 198 ÷ 2 = __ r __

Step 2: divide the quotient by 2. __ ÷ 2 = __ r __

Step 3: repeat until you have 0.

__ ÷ 2 = __ r __

Step 4: write the remainders from last to first.

Answer: 11000110

3 Convert 85₁₀ to hexadecimal (base 16).

Step 1: divide the number by 16. 85 ÷ 16 = 5 r __

Step 2: divide the quotient by 16. 5 ÷ 16 = __ r __

Step 3: repeat until you have 0.

Step 4: write the remainders from last to first, replacing any 2-digit numbers with their letter.

Answer: 55

4 Convert 210₁₀ to hexadecimal.

Step 1: divide the number by 16. 210 ÷ 16 = __ r __

Step 2: divide the quotient by 16. 13 ÷ __ = __ r __

Step 3: repeat until you have 0.

Step 4: write the remainders from last to first, replacing any 2-digit numbers with their letter.

Answer: D2

Independent Practice

Convert 42₁₀ to binary.
Convert 195₁₀ to binary.
Convert 210₁₀ to binary.
Convert 57₁₀ to hexadecimal.
Convert 132₁₀ to hexadecimal.
Convert 200₁₀ to hexadecimal.

Answers

101010
11000011
11010010
39
84
C8