Understanding Non-Base 10 Systems

In base-10, place values correspond to powers of 10, for example:

In base-b, place values correspond to powers of b:

Base-5

Base-5 uses the digits 0, 1, 2, 3, 4 to represent any number. In base-5, we start by counting 1, 2, 3, 4. At 5, however, we must write 10₅ (base-5 does not use the digit “5”):

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Here’s how we write to up to 10 in base-5.

Q1.   Convert 4302₅ to decimal notation.

Q2.   Write 384 as a base-5 numeral.

Q3.   Add 342₅ + 223₅

Q4.   Subtract 424₅ − 143₅

Q5.   Multiply 134₅ × 32₅

Q6.   Perform the division 4132₅ ÷ 23₅

Base-8

The base-8 or octal system uses the digits: 0, 1, 2, 3, 4, 5, 6, 7. Notice that this time the digit “8” isn’t being used. The counting pattern is:

Base-16

The base-16 or hexadecimal system uses the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

The counting pattern is:

Base-2 (Binary)

The base-2 or binary system uses the digits: 0, 1. The counting pattern is:

Q1.   Write the binary command 1011001101011110 using octal notation.

Q2.   Write the binary command 1011001101011110 using hexadecimal notation.