Powers+of+2+-+JL,+JS,+EL

=__**Powers of 2 - THE FACTS!**__=

2^0=1. This is the only odd power. Starting with 2^1, the last digit is periodic with period 4, with the cycle 2 4 8 6. Starting with 2^4, the last two digits are periodic with period 20. The sum of the digits is also periodic starting from 2^6, with the cycle 1, 2, 4, 8, 7, 4.

The first few powers of 2^10 are a little more than those of 1000:
 * 210 = [|1,024]
 * 220 = 1,048,576
 * 230 = 1,073,741,824
 * 240 = 1,099,511,627,776
 * 250 = 1,125,899,906,842,624
 * 260 = 1,152,921,504,606,846,976
 * 270 = 1,180,591,620,717,411,303,424

The number of odd numbers in Pascal's triangle in always a power of 2.

Euclids formula for a perfect number is ((2^k)-1)(2^(k-1)).

__**The Binary Numeral System**__ The **binary numeral system**, or **base-2 number system**, represents numeric values using two symbols, 0 and [|1].

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols **0** through **9**, while binary only uses the symbols **0** and **1**. Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent. For example, the binary number: 100101 is converted to decimal form by: [(**1**) × 25] + [(**0**) × 24] + [(**0**) × 23] + [(**1**) × 22] + [(**0**) × 21] + [(**1**) × 20] = [**1** × 32] + [**0** × 16] + [**0** × 8] + [**1** × 4] + [**0** × 2] + [**1** × 1] = 37
 * __Decimal__ || __Binary__ ||
 * 1 || 1 ||
 * 2 || 10 ||
 * 3 || 11 ||
 * 4 || 100 ||
 * 5 || 101 ||
 * 6 || 110 ||
 * 7 || 111 ||
 * 8 || 1000 ||
 * 9 || 1001 ||
 * 10 || 1011 ||

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 * __Resources- Binary__**

A familiar use of modular arithmetic is in the [|12-hour clock], in which the day is divided into two 12 hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic //modulo// 12.
 * __Modular Arithmetic__**

Modular arithmetic is referenced in [|number theory], [|group theory], [|ring theory], [|knot theory], [|abstract algebra], [|cryptography], [|computer science], [|chemistry] and the [|visual] and [|musical] arts.

Student friendly explanations and examples: []

__Lesson Plans__