​Decimal

What is Decimal?

The Decimal System is how we write numbers.

​We can represent numbers by "expanding" them.

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123 = 1 * 10²​ + 2 * 10¹ + 3 * 10⁰

Decimal

​Decimal

How It Works

Notice the excessive use of tens in decimal representation, since in decimal, we use the ones place, the tens place, the hundreds place, and so on, multiplying every time by ten.​

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Therefore, there is no digit for ten, since​ we have the tens place.

About Decimal

​Binary

What Is Binary?

​123 can be represented as 1 * 10²​ + 2 * 10¹ + 3 * 10⁰

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We can alternatively write it as so.​​

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123 = 1 * 2⁶ + 1 * 2⁵ + 1 * 2⁴​ +​​​​ 1 * 2³​​​ +​ 0 * 2²​​ +​​​ 1 * 2¹​​ +​​ 1 * 2⁰

Binary

​Binary

How It Works

​123 = 1 * 2⁶ + 1 * 2⁵ + 1 * 2⁴​ +​​​​ 1 * 2³​​​ +​ 0 * 2²​​ +​​​ 1 * 2¹​​ +​​ 1 * 2⁰

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Notice how we only multiply the powers of two by 0 or 1.

This is because, if we say ​that 2 = 2 * 2⁰ it is the same thing as 2¹​

So, 123 = 1111011₂​

About Binary

​Binary

Examples

24 = 1 * 2​⁴ + 1 * 2³ + 0 * 2² + 0 * 2¹ + 0 * 2⁰

24 = ​11000₂

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11 = 1 * 2³​ + 0 * 2² + 1 * 2¹ + 1 * 2⁰

11 = ​1011₂

Binary Examples

​Decimal to Binary

Let's turn 45 into binary. We need to divide by 2 until we reach 0.​

43 / 2 = 21 rem 1

21 / 2 = 10 rem 1

10 / 2 = 5 rem 0

5 / 2 = 2 rem 1

2 / 2 = 1 rem 0

1 / 2 = 0 rem 1​

Decimal to Binary

​Now, we list the remainders from the bottom to the top.

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​So 43 in binary is 101011

​Octal and Hexadecimal

Octal and Hexadecimal

Octal is like binary but we have powers of 8, and we can multiply by the digits 0 - 7 rather than just 0 and 1.

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Hexadecimal is similar, but we have powers of 16, and we can multiply by the digits 0 - 15​.

​Hexadecimal Digits

Octal and Hexadecimal

The hexadecimal digits from 0 through 9 are the same as decimal digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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The hexadecimal digits from 10 through 15 are represented via letters.

A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.

AD = 10 * 16¹ + 13 * 16⁰

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​All Bases to Decimal

Turn 101₂ from binary to decimal

101₂ = 1 * 2² + 0 * 2¹ + 1 * 2⁰ = 5​​

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Turn 24₈ from octal to decimal

124​₈ = 1 * 8² + 2 * 8¹ + 4 * 8⁰ = 84​​​

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To decimal

Turn A4₁₆ from hex to decimal​

​A4₁₆​ = 10 (A) * 16¹ + 4 * 16⁰ = ​164

​Which Base?

101₂

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101​₈

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One of these numbers is in binary and one is in octal.​ But they are the same. How do you know which is which? There is a small subscript.

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Important info