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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

2 in domino, dice and card form
Unary - One mark for one number
This is simple counting. One mark is one count.
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

10s
Decimal - Counting 0-9
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

16s
Hexadecimal - 0-F
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) =
= 768 + 48 + 3

8 9 1

Binary - Base2

0,1 The 2 state system
Binary - 0 and 1
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
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