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

81	27	9	3	1
1	2	1	0	0		(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

625	125	25	5	1
4	1	0	0	1		(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 + 21	base 5
(b)	16 + 32	base 7
(c)	141 + 104	base 5
(d)	212 + 121	base 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 × 23	base 5
(b)	122 × 12	base 3
(c)	512 × 24	base 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

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

to see the answer.

(a)	24	to base 3
(b)	16	to base 4
(c)	321	to base 5
(d)	113	to base 6
(e)	314	to base 7
(f)	84	to base 9
(g)	142	to base 3
(h)	617	to base 5

(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

(a)	123	Base:
(b)	112	Base:
(c)	184	Base:

(a)	13 + 23	in base 4
(b)	120 + 314	in base 5
(c)	222 + 102	in base 3
(d)	310 + 132	in base 4
(e)	624 + 136	in base 7
(f)	211 + 142	in base 5
(g)	333 + 323	in base 4
(h)	141 + 424	in base 5

(a)	+ =
(c)	+ =
(e)	+ =

(a)	412	base 5
(b)	333	base 4
(c)	728	base 9
(d)	1210	base 3
(e)	1471	base 8
(f)	612	base 7
(g)	351	base 6
(h)	111	base 3

(a)	3 × 2	in base 4
(b)	4 × 3	in base 5
(c)	4 × 2	in base 6
(d)	3 × 5	in base 6
(e)	2 × 2	in base 3
(f)	8 × 8	in base 9

(a)	121 × 11	in base 3
(b)	133 × 12	in base 4
(c)	13 × 24	in base 5
(d)	142 × 14	in base 5
(e)	161 × 24	in base 7
(f)	472 × 32	in base 8
(g)	414 × 22	in base 5
(h)	2101 × 21	in base 3

(a)	4 + 2 = 11	Base:
(b)	7 + 5 = 13	Base:
(c)	8 × 2 = 17	Base:
(d)	4 × 5 = 32	Base:
(e)	11 – 3 = 5	Base:
(f)	22 – 4 = 13	Base:

(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.

(a)	171 × 12 = 2272	Base:
(b)	122 × 21 = 11102	Base:
(c)	24 × 32 = 1252	Base:
(d)	333 × 33 = 23144	Base: