Barry got almost all of it right.
ㅑ/ㅕ/ㅛ/ㅠ are not double vowels, they are iotized vowels, which means they are really a consonant combined with a vowel (y+vowel), keeping in mind that the linguistic definition of a consonant is where the tongue (or any part of the oral cavity above the larynx) must touch some part of the mouth to pronounce.
ㅐ/ㅒ/ㅔ/ㅖ/ㅢ/ㅚ/ㅘ/ㅙ/ㅟ/ㅝ/ㅞ aren't combined vowels, they are dipthongs
Barry has reversed the Korean-Chinese style (more accurately called Sino-korean) and the pure Korean styles.
일 (一), 이 (二), 삼 (三), 사 (士), 오 (五) is Sino Korean
하나, 둘, 셋, 넷, 다섯 is pure Korean
Koreans also have counting words for days (하루, 이틀), numbers of eggs (줄, 판), number of sheets of laver [seaweed] (퇴, 톳), and probably a lot more too.
There are also 11 consonant clusters ㄳ/ ㄵ/ ㄶ/ ㄺ/ ㄻ/ ㄼ/ ㄽ/ ㄾ/ ㄿ/ ㅀ/ ㅄ