1248 Method: Quick Mental Binary Conversion

The standard method for converting decimal to binary often involves repeated division by 2, which usually requires writing out calculations. Here, we introduce a method designed for mental arithmetic—the "1248 Method"—that allows you to perform these conversions quickly without needing scratch paper.

Binary to Decimal

Let's start with some simple examples:

• 110
• 111
• 1010

Can you quickly determine the decimal values for these three binary numbers without memorizing them?

The decimal values are 6, 7, and 10. How do we calculate this quickly? Look at the table below:

8	4	2	1
	1	1	0

For 110: treating 0 as false and 1 as true, the positions with "true" (1) are 4 and 2. Adding them together: $4 + 2 = 6$. This is the decimal value of 110.

Now apply this to 1010. Based on the 1248 theory, we draw the model:

8	4	2	1
1	0	1	0

The positions with 1 are 8 and 2. Adding them gives $8 + 2 = 10$.

This makes decimal to binary conversion simple. But what about larger numbers, like 101010?

The sequence continues: 1, 2, 4, 8, 16, 32, 64, 128... These are powers of 2.

32	16	8	4	2	1
1	0	1	0	1	0

Sum the numbers where the bit is 1: $32 + 8 + 2 = 42$. The decimal value of 101010 is 42.

Decimal to Binary

Now, let's calculate the binary form for these three numbers:

• 9
• 23
• 77

Using the 1248 method, we decompose the number 9 into powers of 2: $9 = 8 + 1$. Mapping this to the table:

8	4	2	1
1	0	0	1

The "8" and "1" positions are set to 1, and the others are 0. The result is 1001.

For 23, decompose it: $23 = 16 + 7 = 16 + 4 + 2 + 1$.

16	8	4	2	1
1	0	1	1	1

The binary for 23 is 10111.

Similarly, for 77: $77 = 64 + 13 = 64 + 8 + 4 + 1$.

64	32	16	8	4	2	1
1	0	0	1	1	0	1

Filling in the zeros, 77 in binary is 1001101.

With this method, you can easily handle binary-decimal conversions mentally by visualizing the table.

Decimal and Hexadecimal Conversion

It's also worth mentioning decimal to hexadecimal conversion. It's simple but easy to overlook if not practiced.

Calculate the hexadecimal values for:

• 24
• 36
• 82

$$24 = 16 \times 1 + 8 \Rightarrow 0\text{x}18$$

$$36 = 16 \times 2 + 4 \Rightarrow 0\text{x}24$$

$$82 = 16 \times 5 + 2 \Rightarrow 0\text{x}52$$

Hexadecimal numbers are composed of the multiples of 16 plus the remainder.

Hexadecimal to Decimal is the reverse:

$$0\text{x}18 = 1 \times 16^1 + 8 \times 16^0 = 24$$

$$0\text{x}24 = 2 \times 16^1 + 4 \times 16^0 = 36$$

$$0\text{x}52 = 5 \times 16^1 + 2 \times 16^0 = 82$$

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