Number System - Binary, Octal, Hex

Binary, octal, and hexadecimal are all place-value systems. The difference is the base. Binary is base $2$, octal is base $8$, and hexadecimal is base $16$. Once that idea clicks, the symbols stop looking mysterious.

In any positional number system, each place is a power of the base. In base $10$, the places are $1$, $10$, $100$, and so on. In base $2$, the places are $1$, $2$, $4$, $8$, $16$, and so on. The same rule works for every base.

What Each Number System Uses

Binary uses only the digits $0$ and $1$.

Octal uses the digits $0$ through $7$.

Hexadecimal uses $16$ symbols: $0$ through $9$, then $A$ through $F$ for values $10$ through $15$.

That means a hexadecimal digit can hold more information than a binary digit because one hex place counts in powers of $16$, not powers of $2$.

The Main Intuition

A number does not change its value just because you write it in a different base. Only the representation changes.

For example, the base-$10$ number $45$ is still the same quantity whether you write it in binary, octal, or hexadecimal. Different bases are like different languages for the same amount.

One Strong Example: Write $45$ In Binary, Octal, And Hex

Start with base $10$.

$$45 = 32 + 8 + 4 + 1$$

Those are powers of $2$:

$$32 = 2^5,\quad 8 = 2^3,\quad 4 = 2^2,\quad 1 = 2^0$$

So the binary form has $1$s in the $2^5$, $2^3$, $2^2$, and $2^0$ places:

$$45_{10} = 101101_2$$

Now use the binary form to get octal. Since $8 = 2^3$, group binary digits in sets of $3$ from the right:

$$101101_2 = 101\ 101_2$$

Each group becomes one octal digit:

$$101_2 = 5,\quad 101_2 = 5$$

So

$$45_{10} = 55_8$$

Now get hexadecimal. Since $16 = 2^4$, group binary digits in sets of $4$ from the right. Add leading zeros if needed:

$$101101_2 = 0010\ 1101_2$$

Then convert each group:

$$0010_2 = 2,\quad 1101_2 = 13 = D$$

So

$$45_{10} = 2D_{16}$$

All three forms represent the same quantity:

$$45_{10} = 101101_2 = 55_8 = 2D_{16}$$

Common Mistakes

One common mistake is forgetting that the base changes the place values. The string $101$ does not mean the same thing in base $2$, base $8$, and base $10$.

Another mistake is using digits that the base does not allow. For example, $2$ cannot appear in a binary number, and $8$ cannot appear in an octal number.

Students also often group binary digits incorrectly when converting to octal or hexadecimal. Group from the right, and add leading zeros if you need a complete group.

When These Number Systems Are Used

Binary is the basic language of digital systems because switches naturally have two states. Octal and hexadecimal are compact ways to write long binary strings.

You do not need computer science to understand the math idea. These systems are still valuable because they train the core rule behind all positional notation: the value depends on the base and the place.

Try A Similar Conversion

Try converting $58_{10}$ into binary, octal, and hexadecimal. First write it as a sum of powers of $2$, then group the binary digits to get the other two forms.