Simple Captcha Login WordPress

functions.php

//CAPTCHA
// function number to word
function number_to_word($number) {
  $words = array(
      'zero', 'one', 'two', 'three', 'four', 'five', 'six', 'seven', 'eight', 'nine',
      'ten', 'eleven', 'twelve', 'thirteen', 'fourteen', 'fifteen', 'sixteen', 'seventeen', 'eighteen', 'nineteen',
      'twenty', 30 => 'thirty', 40 => 'forty', 50 => 'fifty', 60 => 'sixty', 70 => 'seventy', 80 => 'eighty', 90 => 'ninety'
  );

  if(!is_numeric($number)) {
      return false;
  }

  if(($number >= 0 && $number <= 20) || $number == 30 || $number == 40 || $number == 50 || $number == 60 || $number == 70 || $number == 80 || $number == 90) {
      return $words[$number];
  }

  if($number > 20 && $number < 100) {
      $tens = (int)($number / 10) * 10;
      $ones = $number % 10;
      return $words[$tens] . '-' . $words[$ones];
  }

  return false;
}
// add formula captcha to login page
function add_formula_captcha() {
    // generate two random numbers between 0 and 10 and store them in the session
    session_start();
    $num1 = rand(0, 10);
    $num2 = rand(0, 10);
    $_SESSION['formula_captcha'] = $num1 + $num2;

    // output the captcha to the login page
    echo '<p>Please enter the answer to <strong>' . $num1 . ' + ' . number_to_word($num2) . '</strong>: <input type="text" name="formula_captcha" /></p>';
}

// validate formula captcha on login
function validate_formula_captcha($user, $username, $password) {
    // check if the formula captcha matches the user input
    session_start();
    $formula_captcha = $_SESSION['formula_captcha'];
    $user_captcha = $_POST['formula_captcha'];
    if ($user_captcha != $formula_captcha) {
        // if the captcha doesn't match, prevent the login and display an error message
        remove_action('authenticate', 'wp_authenticate_username_password', 20);
        return new WP_Error('invalid_captcha', 'Invalid formula captcha.');
    }
    return $user;
}
add_action('login_form', 'add_formula_captcha');
add_filter('authenticate', 'validate_formula_captcha', 30, 3);