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While we as humans intuitively use decimal base in our everyday lives, there are other base systems that are very important to mathematics and computers. For example, binary, base 2, is the underlying representation of numbers for all modern computers. However, in this discrete mathematics lesson, we will also look at other bases. To start, let’s look at some popular bases less than or equal to base 10.

• In base 2, binary, there are two numbers per digit (0, 1).
• In base 4, we use 4 numbers per digit (0, 1, 2, 3).
• In base 8, octal, we use 8 numbers per digit (0, 1, 2, 3, 4, 5, 6, 7).
• As we know for base 10, decimal, we use 10 numbers per digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Note how the base number itself never shows up in the list of allowable digits because we start with 0 in the allowable number values for each digit.

The binary base system is part of the backbone of computing. Binary numbers reflect the underlying hardware of computers, where approximately 5 volts = 1 and approximately 0 volts = 0. That is, the software interprets the higher signal as a “1” and the lower signal as “0.” The use of 0 V and 5 V is a standard for nearly every programmable electronic device, from the simplest sensor to the most complex supercomputer.

Binary numbers can also represent Boolean values, which are either true (1) or false (0). Because of the Boolean values, we can also use binary numbers to code logical statements. Specially-designated binary numbers also represent various alphabets, including ASCII and Unicode. ASCII is an older system for representing characters, and Unicode is the current international standard for characters.

### Instructions

1.

If we need to use base seven, list the highest value we can see in any column.

Assign the value to checkpoint_1 in the code editor.

2.

What is the maximum voltage in computer hardware to signal a “1?”

Assign the value to checkpoint_2 in the code editor.