problem_123.js

/**
 * Determines whether string represents number
 * @param  {string}  str
 * @return {boolean}
 */
function isValidNumber(str) {
  const arr = str.trim().split('');

  if (arr.length === 0) return false;

  if (arr.length === 1 && !Number.isInteger(arr[0])) return false;

  // If the 1st char is not '+', '-', '.' or digit
  if (arr[0] !== '+' && arr[0] !== '-' && !isInteger(arr[0]) && arr[0] !== '.')
    return false;

  let flag = false;

  for (let i = 1; i < arr.length; i++) {
    // If any of the char does not belong to [digit, +, -, ., e]
    if (
      !isInteger(arr[i]) &&
      arr[i] !== 'e' &&
      arr[i] !== '.' &&
      arr[i] !== '+' &&
      arr[i] !== '-'
    )
      return false;

    if (arr[i] === '.') {
      // checks if the char 'e' has already occurred before '.'
      if (flag === true) return false;

      // If '.' is the last character.
      if (i + 1 >= arr.length) return false;

      // if '.' is not followed by a digit.
      if (!isInteger(arr[i + 1])) return false;
    } else if (arr[i] === 'e') {
      flag = true;
      // if there is no digit before 'e'.
      if (!isInteger(arr[i - 1])) return false;

      // If 'e' is the last Character
      if (i + 1 >= arr.length) return false;

      // if e is not followed either by '+', '-' or a digit
      if (!isInteger(arr[i + 1]) && arr[i + 1] !== '+' && arr[i + 1] !== '-')
        return false;
    }
  }

  return true;
}

/**
 * Finds string to integer representation
 * @param  {string}  char
 * @return {boolean}
 */
function isInteger(char) {
  return Number.isInteger(parseInt(char, 10));
}

console.log(isValidNumber('10')); // true
console.log(isValidNumber('-10')); // true
console.log(isValidNumber('10.1')); // true
console.log(isValidNumber('-10.1')); // true
console.log(isValidNumber('1e5')); // true

console.log(isValidNumber('a')); // false
console.log(isValidNumber('x 1')); // false
console.log(isValidNumber('a -2')); // false
console.log(isValidNumber('-')); // false