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At this point, it would be possible to change back to real numbers using the formula e 2ix + e −2ix = 2 cos 2x. Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:
if a number is divisible by 2, but not by 3, its square ends in 4, and its preceding digit must be 0, 1, 4, 5, 8, or 9; and; if a number is not divisible by 2, but by 3, its square ends in 9, and its preceding digit must be 0 or 6. Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example).
Sri Sultan Hamengkubuwono IX (Hanacaraka: ꦯꦿꦶꦯꦸꦭ꧀ꦡꦟ꧀ꦲꦩꦼꦁꦑꦸꦨꦸꦮꦟ꧇꧙꧇; 12 April 1912 – 2 October 1988 [a]), often abbreviated as HB IX, was an Indonesian politician and Javanese royal who was the second vice president of Indonesia, the ninth sultan of Yogyakarta, and the first governor of the Special Region of Yogyakarta.