Why does an A at 440 hertz have octaves at 220 and 880 hertz?

In music, an octave is defined as an interval where the higher note has a sound-wave frequency exactly twice that of the lower note, resulting in a frequency ratio of 1:2.12

Because of this 1:2 ratio, the octave is considered an acoustically pure interval. When a reference note like A is set to 440 hertz, the octave above it is calculated by multiplying the frequency by 2 ($440\times 2=880$ Hz), and the octave below it is calculated by dividing the frequency by 2 ($440÷2=220$ Hz).31

Due to this close acoustic relationship, the upper and lower octaves are perceived as qualitatively identical to the reference note, despite the difference in pitch.1