Positional Notation

We use positional notation to represent numbers

positional notation¶

A numeral system in which each digit position has a place value, and the value of a number is the sum of each of the digits times its place value.

In positional notation we must define our numeral system's base - how much each position is worth.

This is the total number of unique numeral symbols we use, including 0.

In our familiar base 10, each place is worth a power of 10:

 12   - 1 times 10 plus 2 times 1
404   - 4 times 100 plus 0 times 10 plus 4 times 1

In base 2, there are only two symbols, 0 and 1, so each place is worth a power of 2.

  1   - 1 * 1                             [1, in base 10]
 10   - (1 * 2) plus (0 * 1)              [2]
101   - (1 * 4) plus (0 * 2) plus (1 * 1) [5]

Numerals¶

We're all familiar with the numerals of our base-10 system, in which each place is worth a power of 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

When we count past 9 we put a 1 in the next higher place and a 0 in the first place.

For base 2 counting we need only two numeral symbols, or digits: 0 and 1.

11001   - (1 * 16) + (1 * 8) + (0 * 4) + (0 * 2) + (1 * 1) [25]

Note that any leading zeros don't change the overall value:

00011001   - (0*128) + (0*64) + (0*32) + (1*16) + (1*8) + (0*4) + (0*2) + (1*1) [still 25]

The more symbols our numeral system has, the less places it takes to represent a large number.

Drill¶

Write out the following decimal numbers in binary: * 3 * 4 * 7 * 11 * 16 * 15 * 20 * 21

Solution: NumeralSystems/com.example.numeralsystems.solutions/SkillDrills1.java

Prev -- Up -- Next