4.05.md

Section 4.05: Numerical Systems

Let's talk about numberical systems including hexadecimal and going from decimal to binary, etc.

The numeral system we typically use in our day to day lives is base-ten or decimal. Why do we use base-ten? Probably because we have 10 fingers.

	ten millions	millions	hundred thousands	ten thousands	thousands	hundreds	tens	ones
decimal	107	106	105	104	103	102	101	100

	128s	sixty fours	thirty twos	sixteens	eights	fours	twos	ones
binary	27	26	25	24	23	22	21	20

				65,536s	4,096s	256s	16s	ones
hexadecimal	167	166	165	164	163	162	161	160

|||||262144's|4096's|64's|ones| |------------|------------|--------|-----------------|-------------|---------|--------|----|----| |base64|64⁷|64⁶|64⁵|64⁴|64³|64²|64¹|64⁰|

In the table, you can see we have ones, tens, hundreds, thousands, etc. In each spot for the decimal system, we can put 10 digits, which are 0 through 9. So, for example when we have 9 ones and add 1, we then have 1 ten and 0 ones, represented as 10. If I said I had twelve dollars, that's a 1 in the tens spot and a 2 in the twos spot. Reviewing these principles will help in understanding how binary and hexadecimal work.

So, how does the decimal (or base-ten) system work? We can put 1 digit in each spot, and it can be one of 10 values from 0 through 9. Likewise, the binary (or base-two) system has 1 digit in each spot, which can be one of 2 values, from 0 through 1, and the hexadecimal (or base-sixteen) system can also has 1 digit in each spot, which can be one of 16 values.

Since 10 through 15 are actually 2 characters, we use a through e instead, so the sixteen values are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e.

So, we can use binary and hexidecimal to represent numbers, just like we can with decimal, though we have different amounts of values we can store in each digit.

Let's look at what some numbers look like in the various systems

decimal	binary	hexadecimal
0	0	0
1	1	1
5	101	5
10	1010	a
15	1111	e
50	110010	32
100	1100100	64

decimal to hexadecimal and binary conversion calculator

Here's a binary joke, "There are 10 types of people in the world; those who understand binary and those who don't."

To represent 42,420 we need 4 ten thousands, 2 thousands, 4 hundreds, and 2 tens:

ten millions	millions	hundred thousands	ten thousands	thousands	hundreds	tens	ones
107	106	105	104	103	102	101	100
			4	2	4	2	0

How would we represent 42 in binary?

128's	sixty fours	thirty twos	sixteens	eights	fours	twos	ones
27	26	25	24	23	22	21	20
		1	0	1	0	1	0

And how about representing the number 911 in hexadecimal

65,536s	4,096s	256s	16s	ones
164	163	162	161	160
		3	8	F

To get 911 in base-sixteen, we have 3 times 256 which equals 768 plus 8 times 16 which equals 128 and 15 (represented by F) times 1 which equals 15. 768 + 128 + 15 = 911.

So now that we know how decimal, binary, and hexadecimal work, let's see it in action in Go:

package main

import (
	"fmt"
)

func main() {
	s := "H"
	fmt.Println(s)
	
	/* convert s to a slice of bytes */
	bs := []byte(s)
	fmt.Println(bs)
	
	/* assign n the value of the first slice of bytes in bs */
	n := bs[0]
	fmt.Println(n)
	
	/* print the type of n */
	fmt.Printf("%T\n", n)
	
	/* print n as binary */
	fmt.Printf("%b\n", n)
	
	/* print n as hexadecimal */
	fmt.Printf("%x\n", n)
	
	/* print n as hexadecimal with indicator prefix */
	fmt.Printf("%#X\n", n)
}

playground