Mr Zuma said “the nation has lost its greatest son”, adding “he is now resting. He is now at peace.”
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He died surrounded by his family at his Johannesburg home at 8.50pm local time.
Mr Mandela, who was 95, will get a state funeral and national flags will be lowered to half mast, said Mr Zuma.
He had been receiving medical treatment for the last three years and for the last six months had been critically ill.