"how numbers are stored and used in computers"
Humans represent numbers in base 10 - that is, with 10 different digits.
Try clicking the digits below to update the formula.
Each digit corresponds to a power of 10. Humans also occassionally write numbers in base 16, where the letters a through f represent the numbers 10 through 15. This is commonly seen in web programming as a way to specify a color.
Click the digits or change the color to update the hex code.
Regardless of which human representation you choose, a computer will represent that number in base 2, with only the digits 0 and 1. In computer hardware, this corresponds to the presence or absence of an electrical current.
Each digit of a binary number is referred to as a bit. You will often see binary numbers described in terms of their bit length, such as the 8 bit integer below.
We call it an integer because it can only store whole numbers. Since there is no way to specify a negative number, we would also call this an unsigned integer.
A group of 8 bits is called a byte, where each byte can represent 28 (256) different values. By combining multiple bytes, a number can have a greater range of possible values.
In the example above, we are using 8 bits to represent an unsigned integer. If we want to store negative numbers, we can dedicate 1 bit as a sign bit to store 0 for positive numbers and 1 for negative numbers.
If the sign bit simply controlled the presence or absence of a negative sign (-), there would be two binary numbers that both represent zero -
To avoid ambiguity, and to allow the same circuits to be used for both signed and unsigned numbers, computers represent negative numbers in two's complement. If the sign bit is 1, then all bits are inverted, so 1 is considered a 0, and 0 is considered a 1 - the sum is then incremented by 1, and the result is multiplied by -1 to get the negative value.
Just as we express long distances with kilometers instead of meters, we can express large amounts of binary information with prefixes that indicate their order of magnitude. Unlike traditional scientific prefixes which are expressed in powers of 10 (e.g. 1 kilometer is 1000 meters), binary prefixes often refer to the closest power of 2, so a kilobyte is technically 1024 bytes (
Here is what one kilobyte (1024 bytes) of information looks like. This isn't just any kilobyte though - it is De rerum natura, the ancient wisdom of Lucretius.
Mother of Rome, delight of Gods and men, Dear Venus that beneath the gliding stars Makest to teem the many-voyaged main And fruitful lands- for all of living things Through thee alone are evermore conceived, Through thee are risen to visit the great sun- Before thee, Goddess, and thy coming on, Flee stormy wind and massy cloud away, For thee the daedal Earth bears scented flowers, For thee waters of the unvexed deep Smile, and the hollows of the serene sky Glow with diffused radiance for thee! For soon as comes the springtime face of day, And procreant gales blow from the West unbarred, First fowls of air, smit to the heart by thee, Foretoken thy approach, O thou Divine, And leap the wild herds round the happy fields Or swim the bounding torrents. Thus amain, Seized with the spell, all creatures follow thee Whithersoever thou walkest forth to lead, And thence through seas and mountains and swift streams, Through leafy homes of birds and greening plains, Kindling the lure of love in every breast, Thou bringest the eternal generations forth, Kind after kind. And since 'tis thou alone Guidest the Cosmos, and without thee naught Is risen to reach the shining shores of light, Nor aught of joyful or of lovely born, Thee do I crave co-partner in that verse Which I presume on Nature to compose For Memmius mine, whom thou hast willed to be Peerless in every grace at every hour- Wherefore indeed, Divine one, give my words Immortal charm. Lull to a timely rest O'er sea and land the savage works of war, For thou alone hast power with public peace To aid mortality; since he who rules The savage works of battle, puissant Mars, How often to thy bosom flings his strength O'ermastered by the eternal wound of love- And there, with eyes and full throat backward thrown, Gazing, my Goddess, open-mouthed at thee, Pastures on love his greedy sight, his breath Hanging upon thy lips. Him thus reclined Fill with thy holy body, round, above! Pour from those lips soft syllables to win Peace for the Romans, glorious Lady, peace! For in a season troublous to the state Neither may I attend this task of mine With thought untroubled, nor mid such events The illustrious scion of the Memmian house Neglect the civic cause.
Each character of Lucretius' poem is encoded with a single byte, using a format called ASCII. This format works well for English letters, but is woefully inadequate for modern text which contains a wide range of characters like emojis and Chinese letters. Most modern systems use encoding formats like UTF-8 and UTF-16, which can use multiple bytes to encode each character.