Hong Kong
Stage 1 - Stage 3

# Numbers with Different Bases

Lesson

The base of a number system determines entirely how it operates.  Our normal number system (the decimal system) uses a base $10$10

Each of our place value columns is $10$10 times the size of the previous columns.  So $10$10 ones, yields $1$1 ten, and $10$10 tens yield $1$1 hundred and so on.  Each column can be represented as powers of $10$10.

ten thousands thousands hundreds tens units
$10^4$104 $10^3$103 $10^2$102 $10^1$101 $10^0$100

However, number systems don't always have to be in base $10$10

## Base-2: Binary

A base-$2$2 number system is covered in detail here. The binary number system, or base-$2$2 number system, represents numerical values using two different symbols: $0$0 and $1$1. This system is used internally in most computers and computer-based devices, such as mobile phones.

The binary number system uses powers of $2$2, instead of powers of $10$10 to construct the place value columns.

sixty-fours thirty-twos sixteens eights fours twos ones
$2^6$26 $2^5$25 $2^4$24 $2^3$23 $2^2$22 $2^1$21 $2^0$20

So in binary form a number written like

$1011$1011, actually means $1\times8+1\times2+1\times1$1×8+1×2+1×1 which in decimal is $11$11.

Going the other way now let's write the number $29$29 in binary.

$29$29 is comprised of $1\times16+1\times8+1\times4+0\times2+1\times1$1×16+1×8+1×4+0×2+1×1 which we would write as $11101$11101.

## Base-3: Ternary

The ternary number system, or base-$3$3 number system, represents numerical values using three different symbols: $0$0 and $1$1 and $2$2.  This system is not used as commonplace as binary, but a ternary like system is used both in baseball scoring (keeping track of innings and outs) and in Islam when keeping track of prayers using Tasbih.

The ternary number system uses powers of $3$3, instead of powers of $2$2 like binary, or $10$10 like decimal to construct the place value columns.

two hundred and forty threes eighty-ones twenty-sevens nines threes ones
$3^5$35 $3^4$34 $3^3$33 $3^2$32 $3^1$31 $3^0$30

So in ternary form a number written like

$2021$2021, actually means $2\times27+2\times3+1\times1$2×27+2×3+1×1 which in decimal is $61$61

Going the other way now let's write the number $29$29 in ternary.

$29$29 is comprised of $1\times27+0\times9+0\times3+2\times1$1×27+0×9+0×3+2×1 which we would write as $1002$1002

## Base-8: Octal

The octal number system, or base-$8$8 number system, represents numerical values using eight different symbols: $0$0 ,$1,2,3,4,5,6$1,2,3,4,5,6 and $7$7. This system is often used as a focus for investigations of numbers in other bases due to popular cartoon character lands (where they only have $8$8 fingers).

The octal number system uses powers of $8$8, instead of powers of $10$10 to construct the place value columns.

four thousand and ninety sixes five hundred and twelves sixty-fours eight ones
$8^4$84 $8^3$83 $8^2$82 $8^1$81 $8^0$80

So in octal form a number written like

$2031$2031, actually means $2\times512+3\times8+1\times1$2×512+3×8+1×1 which in decimal is $1049$1049

Going the other way now let's write the number $29$29 in octal.

$29$29 is comprised of $3\times8+5\times1$3×8+5×1 which we would write as $35$35.

The hexadecimal number system, or base-$16$16 number system, represents numerical values using $16$16 different symbols: $0$0 ,$1,2,3,4,5,6,7,8,9$1,2,3,4,5,6,7,8,9 and then the letters $A,B,C,D,E,F$A,B,C,D,E,F. This system is often widely in digital images and digital design as it formulates the bases of the RBG (red, blue, green) colour system used to get combinations of these primary colours and shades of colour.

In the RGB Colour System used in computers there are $256$256 possible levels of brightness for each component in an RGB (Red, Green, Blue) colour. So the red component, blue component and the green component can each be represented by two hexadecimal digits. An entire RGB colour can therefore be represented by six hexadecimal digits. Black is depicted as no colour #000000 and white is depicted as all colours #FFFFFF.

The hexadecimal number system uses powers of $16$16, instead of powers of $10$10 to construct the place value columns.

two hundred and fifty sixes sixteens ones
$16^2$162 $16^1$161 $16^0$160

So in hexadecimal form a number written like

$2E9$2E9, actually means $2\times256+E\times16+9\times1=2\times256+14\times16+9\times1$2×256+E×16+9×1=2×256+14×16+9×1 which in decimal is $745$745

Going the other way now let's write the number $29$29 in hexadecimal.

$29$29 is comprised of $1\times16+13\times1$1×16+13×1 which we would write as $1D$1D

#### Worked examples

##### Question 1

What does the $5$5 represent in $234_5$2345?

1. Base

A

Power

B

Numeral

C

Positional Value

D

##### Question 2

Convert $43$43 in base-16 into base-10.

##### Question 3

Consider $30$30 in base-10.

1. Convert $30$30 to base-16 by filling in the boxes.

Base-10 Base-16 $\editable{3}$3 $\editable{0}$0 $\editable{}$ $\editable{}$