private static int[] sub(int[] sub1, int[] sub2) {
		if (sub2.length == 1 && sub2[0] == 0) return sub1;
		int[] result = new int[sub2.length];
		for (int i = 0; i < sub2.length; i++){
			if (sub2[i] == 1) result[i] = 0;
			else result[i] = 1;
		}
		int[] one = {1};
		result = add(result, one);
		result = add(sub1, result);
		if (result[result.length - 1] == 1) result[result.length - 1] = 0;
		return trimZeroes(result);
	}
private static int[] trimZeroes(int[] toTrim){
		if (toTrim[toTrim.length - 1] == 0) {
			int firstOne = 0;
			boolean firstOneReached = false;
			for (int i = toTrim.length - 1; !firstOneReached && i >= 0; i--){
				if (toTrim[i] == 1){
					firstOne = i;
					firstOneReached = true;
				}
			}
			int[] result = new int[firstOne + 1];
			for (int i = 0; i < result.length; i++){
				result[i] = toTrim[i];
			}
			return result;
		} else return toTrim;
	}
private static int[] add(int[] addend1, int[] addend2) {
		int[] result = new int[Math.max(addend1.length, addend2.length) + 1];
		for (int i = 0; i < Math.min(addend1.length, addend2.length); i++){
			result[i] = addend1[i] + addend2[i];
		}
		if (addend1.length == addend2.length) {
			return carryCorrect(result);
		} else if (addend1.length > addend2.length) {
			result[addend1.length - 1] = addend1[addend1.length - 1];
			return carryCorrect(result);
		} else {
			result[addend2.length - 1] = addend1[addend2.length - 1];
			return carryCorrect(result);
		}
	}
private static int[] carryCorrect(int[] toCorrect) {
		// we're starting at 1 because the first digit will never need correction
		for (int i = 0; i < toCorrect.length; i++){		
			// affecting i+1 is normally dangerous as it can lead to an index out of bounds exception
			// except here we know that most significant digit won't ever trigger these if-statements
			if (toCorrect[i] > 1) {
				// if the entry is an even number, we know that this should be a binary 0
				if (toCorrect[i]%2 == 0) {	
					toCorrect[i+1] += toCorrect[i]/2; // carry the total complete bits to the next column
					toCorrect[i] = 0;
				// otherwise the entry is an odd number, so 
				} else {
					toCorrect[i+1] += (toCorrect[i] - 1)/2; // carry the total complete bits to the next column
					toCorrect[i] = 1;
				}
			}
		}
		
		return trimZeroes(toCorrect);
	}