Our numbering system consists of 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) with which we can write any number. This system is called decimal base. We can demonstrate any number by applying powers of base 10. Look:
47 621 = 4 x 104 + 7 x 10³ + 6 x 10² + 2 x 10¹ + 1 x 100
In the binary system (base 2), numbers are written using 0 and 1 and powers of 2.
Look:
21(base 10) = 10101(base 2)
Transforming the number in base 2 to base 10. Watch:
10101(base 2) = 1 x 24 + 0 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 20 (according to the position of 0 and 1 we raise base 2 to the proper exponent)
| 10101 | Base 2 |
| 1 | Position 4: 24 |
| 0 | Position 3: 2³ |
| 1 | Position 2: 2² |
| 0 | Position 1: 2¹ |
| 1 | Position 0: 20 |
1 x 24 + 0 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 20
16 + 0 + 4 + 0 + 1
21
In the barcode, the black bar means 1 and the white one means 0. The number 10100001101, in base 2 is a simplified barcode.
The prices of products in commerce shown in barcode are more complex, but we can establish the barcode below in base 10.
10100001101base2 = 1 x 210 + 0 x 2
9 + 1 x 28 + 0 x 27 + 0 x 26 + 0 x 25 + 0 x 24 + 1 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 201 x 1024 + 0 x 512 + 1 x 256 + 0 x 128 + 0 x 64 + 0 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1
1024 + 0 + 256 + 0 + 0 + 0 + 0 + 8 + 4 + 0 + 1
1293
The number 1293 is the code for a particular product. The cashier's barcode reader optically scans the codes and in fractions of seconds performs the calculations, reaching the number 1293. Accessing a list of products, look for which was registered with this number, recording the value on the coupon.
by Mark Noah
Graduated in Mathematics
Brazil School Team
Curiosities - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/curiosidades/o-codigo-barras.htm