Unit 1 Section 4 : Other Bases

The ideas that we have considered can be extended to other number bases.

The table lists the digits used in some other number bases.
Base Digits Used
2 0, 1
3 0, 1, 2
4 0, 1, 2, 3
5 0, 1, 2, 3, 4

The powers of the base number give the place values when you convert to base 10.
For example, for base 3, the place values are the powers of 3, i.e. 1, 3, 9, 27, 81, 243, etc.

This is shown in the following example, which also shows how the base 3 number 12100 is equivalent to the base 10 number 144.

Base 3
8127931
12100 (1 × 81) + (2 × 27) + (1 × 9) + (0 × 3) + (0 × 1) = 144 in base 10

The following example shows a conversion from base 5 to base 10 using the powers of 5 as place values.

Base 5
6251252551
41001 (4 × 625) + (1 × 125) + (0 × 25) + (0 × 5) + (1 × 1) = 2626 in base 10

Example 1

Convert each of the following numbers to base 10:

(a)412 in base 6
(b)374 in base 9
(c)1432 in base 5

Example 2

Convert each of the following base 10 numbers to the base stated:

(a)472 to base 4
(b)179 to base 7
(c)342 to base 3

Example 3

Carry out each of the following calculations in the base stated:

(a)14 + 21base 5
(b)16 + 32base 7
(c)141 + 104base 5
(d)212 + 121base 3
Check your answer in (a) by changing to base 10 numbers.

Example 4

Carry out each of the following multiplications in the base stated:

(a)141 × 23base 5
(b)122 × 12base 3
(c)512 × 24base 6
Check your answer to (b) by converting to base 10 numbers.

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.
Question 1
Convert the following numbers from the base stated to base 10:
(a) 412base 5
(b) 333base 4
(c) 728base 9
(d) 1210base 3
(e) 1471base 8
(f) 612base 7
(g) 351base 6
(h) 111base 3
Question 2
Convert the following numbers from base 10 to the base stated:
(a) 24to base 3
(b) 16to base 4
(c) 321to base 5
(d) 113to base 6
(e) 314to base 7
(f) 84to base 9
(g) 142to base 3
(h) 617to base 5
Question 3
Carry out the following additions in the base stated:
(a) 3 + 2 in base 4
(b) 5 + 8 in base 9
(c) 4 + 6 in base 8
(d) 2 + 2 in base 3
(e) 6 + 7 in base 9
(f) 3 + 4 in base 6
Question 4
In what number bases could each of the following numbers be written:
(a) 123 Base:
(b) 112 Base:
(c) 184 Base:
Question 5
Carry out each of the following calculations in the base stated:
(a) 13 + 23in base 4
(b) 120 + 314in base 5
(c) 222 + 102in base 3
(d) 310 + 132in base 4
(e) 624 + 136in base 7
(f) 211 + 142in base 5
(g) 333 + 323in base 4
(h) 141 + 424in base 5
Check your answers to parts (a), (c) and (e) by converting to base 10 numbers.
(a) + =
(c) + =
(e) + =
Question 6
Carry out each of the following multiplications in the base stated:
(a) 3 × 2in base 4
(b) 4 × 3in base 5
(c) 4 × 2in base 6
(d) 3 × 5in base 6
(e) 2 × 2in base 3
(f) 8 × 8in base 9
Question 7
Carry out each of the following multiplications in the base stated:
(a)121 × 11in base 3
(b)133 × 12in base 4
(c)13 × 24in base 5
(d)142 × 14in base 5
(e)161 × 24in base 7
(f)472 × 32in base 8
(g)414 × 22in base 5
(h)2101 × 21in base 3
Check your answers to parts (a), (c) and (e) by converting to base 10 numbers.
(a) × =
(c) × =
(e) × =
Question 8
In which base was each of the following calculations carried out?
(a) 4 + 2 = 11Base:
(b) 7 + 5 = 13Base:
(c) 8 × 2 = 17Base:
(d) 4 × 5 = 32Base:
(e) 11 – 3 = 5Base:
(f) 22 – 4 = 13Base:
Question 9
(a) Change 147 in base 8 into a base 3 number.
(b) Change 321 in base 4 into a base 7 number.
(c) Change 172 in base 9 into a base 4 number.
(d) Change 324 in base 5 into a base 6 number.
Question 10
In which base was each of the following calculations carried out?
(a) 171 × 12 = 2272Base:
(b) 122 × 21 = 11102Base:
(c) 24 × 32 = 1252Base:
(d) 333 × 33 = 23144Base: